Twilight Princess was a much-hyped game in its day, and it remains a polarizing entry in the Zelda series. Far too many people have reviewed this game, and no one is clamoring for another review. But that's OK - I'm going to inflict another review on the world!
SPOILER WARNING
Be forewarned that this post is crawling with spoilers.
Gameplay
Twilight Princess's gameplay builds heavily on Ocarina of Time and the other 3-D Zeldas. That's actually a very firm foundation, and Twilight Princess gives the traditional Zelda mechanics a little bit extra smoothness and responsiveness. Then it throws in some cool new swordplay moves and the opportunity to play as a wolf. I didn't mind Wolf Link, although Link's lupine form could occasionally be clunky in tight areas (especially when dealing with Poes).
Twilight Princess also takes Link's trusted steed to the next level. Epona is a lot more involved in the adventure than in any previous Zelda games. And in an overworld as expansive as Twilight Princess's, you'll really need her to get around - for a while. About midway through the game, you're given the ability to warp almost anywhere you please, which makes Epona a lot less important.
Speaking of big worlds, one of the most common criticisms of Twilight Princess is that its overworld is big but empty. I gotta disagree with this particular complaint. There's a lot to see in almost every corner of the map - Pieces of Heart, Golden Bugs, Poes, or just random caves to discover and explore. If the world feels empty, maybe you need to slow down and look a little more closely...
Twilight Princess gives you a LOT to do. I spent close to 60 hours in my first playthrough - that's something I can say about very, very, very few video games. I'm sure I could've finished faster if I'd followed a walkthrough or spent less time wandering around Hyrule Field and chatting with folks in the towns, but I was in no rush to finish as quickly as possible.
Another oft-cited complaint is the tendency for dungeon items to be one-hit wonders - you use them extensively in the dungeon where you find them, but they don't get much love later in the game. And yeah, that's sometimes the case. Some interesting ideas aren't used much, either. For example, one of the very first things you do in the game is blow on some grass to summon a hawk, which you can use to recover a baby cradle that was stolen. However, this is the only time in the entire game that you use a hawk to retrieve an item (though, to be fair, you quickly obtain items that can pull in items).

Just one of many interesting dungeons in Twilight Princess.
All of the dungeons are interesting and enjoyable in their own way. I didn't find any of the puzzles or combat to be particularly difficult, though. I found it odd that the dungeons tend to get progressively shorter as the game goes on.

The boss of the first dungeon... Zelda's answer to Petey Piranha.
Twilight Princess boasts some really impressive boss battles, although they lean toward the easy side - in most cases, they seem to end almost as soon as you figure out the boss's weak point. I actually found the mini-boss of the second dungeon to be the hardest fight in the game - that was the only time I ever saw the Game Over screen, and one of only a handful of times I glanced at a guide during the main quest.
There are a few quality of life annoyances, many of which were addressed in the HD remake. For a game that hands out Rupees like candy, the wallets are on the small side, which limits the value of all the Orange Rupees given to you by Agitha (and also found in caves and grottoes). Similarly, the Poe hunt can be annoying because unlike the Gold Skulltula quests from OoT, you don't get a visual representation of where you've found all the Poes. It also would've been handy to have a quicker way to change from day to night (or vice versa), since overworld Poes can only be found at night.
Story/Characters

One of many memorable moments...
It's been a number of years since it was fashionable to describe things as "epic" - but Twilight Princess is a Big Epic Zelda Game! Twilight Princess takes a more cinematic approach to the series - there are some very visually striking scenes! Fortunately, it's not one of those modern "movies disguised as video games" - gameplay never takes a back seat to cinema.
Twilight Princess has a LOT of nods to previous Zelda games, particularly Ocarina of Time. At times it can feel a bit derivative, although at least it had the good sense to imitate and expand on one of the best! Twilight Princess is a logical extension of the style of the N64 games, but with the advantage of much more advanced graphical and technical capabilities.
You'll also pick up some A Link to the Past vibes - the Light World/Twilight Realm duality is similar to the Light World/Dark World concept of the third Zelda game. I actually found the "twisted parallel dimension" concept of A Link to the Past to be more interesting than the "this world, but dimly-lit" of Twilight Princess, although I can also see how the Twilight Realm would be more feasible to implement in a 3-D game.
Twilight Princess aimed to be a much darker entry in the series. This was a reaction against Wind Waker, which had bright cel-shaded graphics and lighthearted vibe - and disappointing sales in the Western world. The early trailers certainly conveyed the sense that this game was SERIOUS BUSINESS - that you've been plunged into a Hyrule of darkness and despair. The game certainly has an intense cutscene or two, and some relatively scary monsters, but in reality, it's lighter in tone than Majora's Mask (or even Ocarina of Time, which has some heavy themes and some high-octane nightmare fuel). For what it's worth, Twilight Princess was the first Zelda game to get a "T" ESRB rating, although I missed out on the "animated blood" that it ostensibly contains!
Twilight Princess starts out very linear - you're on rails for most of the first three chapters of the game. Then things open up and you're finally able to start most of the game's sidequests. I actually found the first three chapters to be the most compelling part of the game - you're constantly solving some rather pressing problems in the world, bouncing from one troubled locale to another, often while veiled in a wall of twilight.
The next four chapters focus on exploring random ancient buildings in varying states of disrepair while in search of the four Mirror Shards. There's less padding from things like the Vessel of Light searches, so I can see why some people might prefer this part of the game. However, I find the sense of urgency fades around the time you get the Master Sword. By then, the world seems pretty fine and dandy - the twilight is gone, and the only remaining problem is that massive pyramid-shaped barrier surrounding Hyrule Castle. This is indeed a problem, but almost everyone seems oblivious or indifferent to it. (If Ganondorf is the Great Big King of Evil, he's really slacking at his job.)
Link is a silent protagonist, as typical of the series, but he also enjoys a highly expressive face in this game. It's a more grown-up version of Link, but he's not as stoic as his Ocarina of Time predecessor. He also lives a lifestyle a bit more grounded in reality - he's a farmhand with a real job and a more plausible girlfriend (well, until she forgets who she is, but more on that later). Link's younger neighbors look up to him as a role model, which is cool.
Midna's character has generally been praised - and rightfully so. When you first meet her, she's a selfish jerk with a condescending attitude. If it was somehow possible to enter the game completely unspoiled, you might even question whether she's on your side. She'll likely grow on you, though. As time goes on, though, she softens as a person - and also awakens some pretty spiffy powers of her own! She far transcends the usual Zelda "buddy character" role and actually becomes the primary driver of the game's plot. From a storytelling perspective, it's usually better for the main character to have a bigger role in steering the story instead of constantly chasing another character, but I think Twilight Princess handles this quite nicely.
Zelda's character is an interesting take on the traditional Zelda - you first meet her when she's shrouded in a dark cloak. The interaction between Zelda and Midna is interesting, and while Midna's animosity and resentment toward Zelda is not unwarranted, Zelda proves herself to be a kind and selfless person.
Zant is an excellent villain. He also gets a really fun six-phase boss battle that revisits almost every previous dungeon while you watch him grow increasingly desperate and unhinged!
Ganondorf makes his contractually-obligated appearance as the ultimate villain. Ganondorf makes for an outstanding final battle. On the other hand, Ganondorf feels underdeveloped and a bit tacked-on as a character. He gets less character development than some relatively insignificant characters like Colin!

The young and the restless.
I wanted to like Ilia. (She bears a passing resemblance to a younger version of a certain cute girl I know in real life!) Ilia's not a bad character, but sadly, she gets sent through a series of plot holes and hopelessly-abused clichés. At the beginning of the game, Ilia gets kidnapped by a gang of monsters. We later find out that she was taken to an (almost) abandoned Wild West-styled outpost in the far north of Hyrule, but the game never explains why the baddies go to great lengths to kidnap her and take her to the opposite end of Hyrule. Then she gets amnesia. Classic soap opera amnesia - forgetting her name, forgetting about all her loved ones, yet otherwise being able to function. While both short-term and long-term memory loss are real things, and getting shot in the head can certainly cause neurological issues, this isn't a remotely realistic depiction of amnesia! Even if it was, it's a hopelessly-overused theme in media - something that might work in a Paper Mario game poking fun at the trope, but not so well in a Zelda game. Speaking of amnesia, the story kind of forgets about Ilia for almost half the game, which doesn't help. She may secretly be the most powerful character in the game, though - the game implies that she escapes her confinement in the Hidden Village and makes her way across Hyrule, traversing the (relatively speaking) most dangerous part of the map while alone, in an amnesiac state, and possibly unarmed - and barefoot. The conclusion to Ilia's saga is sweet, but overall, a few tweaks to her arc would've increased both its plausibility and its emotional punch.
Even in a world of twilight, Twilight Princess has its share of goofballs - the mailman, Falbi, Fanadi, Malo, and Agitha all come to mind quickly. And there's a nice cast of well-written supporting characters like Telma, Renado, and Mayor Bo.
As typical of Zelda games, Twilight Princess has no reservations about contradicting previous Zelda lore! (My favorite example is moving the Temple of Time from Hyrule Castle Town to the Sacred Grove, but there are many other examples as well.) This is problematic if you really like the idea of Zelda having a coherent timeline, though it's also pretty clear that the series emphasizes gameplay over continuity.
Graphics

Well, it certainly looks different from Wind Waker!
I can't say Twilight Princess's story is the darkest in the series, but its graphics probably are! As soon as you start the game, you receive a reminder that you might want to adjust your TV's brightness settings. This reminder comes for a reason - Twilight Princess has some very dimly-lit areas, hence its name.

Even Twilight Princess sometimes leaves the twilight behind.
Still, places like Hyrule Field at daytime shine bright. Twilight Princess isn't all gray and brown and grungy like a lot of games/movies/TV shows since this era.
Twilight Princess's graphics have an impressive level of detail by GameCube standards, though there are a few textures that are on the blurry side, and a few models that could benefit from a bit more detail. Twilight Princess has a unique aesthetic that I enjoyed, although it's also true that Wind Waker's graphics have aged a bit more gracefully, despite Wind Waker being closer to the launch of the GameCube.
Character designs are certainly interesting. Nobody's going to accuse the NPCs of all looking the same - even if some of them don't look the way we expect supermodels to look!
Music and Sound
Twilight Princess has an excellent and varied soundtrack. There's a lot of memorable new songs - Ordon Village, Faron Woods, Hyrule Field, Hyrule Castle Town, Lake Hylia, the Light Spirit's theme, the post-boss music, and the Hidden Village all come to mind quickly. A lot of classic Zelda songs are remixed as well - Kakariko Village, Saria's Song, Serenade of Water, and others.

This town ain't big enough for the 21 of you! One of many gems in this soundtrack.
There's an impressive variety of music. For example, most of the bosses get their own theme song, which is nice. And there's lots of areas that easily could've gotten away with recycling a song from another part of the game, but instead they opted to give it a unique tune, which is also nice - Renado's House and Lake Hylia come to mind.
The game's music is MIDI-based, although the soundfonts are reasonably high-quality. A fully orchestrated soundtrack would've been sweet, but the MIDIs end up working well.
Overall, the soundtrack is not as catchy as Ocarina of Time, and doesn't have quite the emotional impact of Majora's Mask, but it's great for what it is. Twilight Princess's soundtrack leans more toward the ambient and the melancholic, which makes sense when you're navigating a world of twilight.
Twilight Princess doesn't have any voice acting. Some people complained about that, although that wasn't a big deal to me.
Sound effects are consistently solid.
Closing Thoughts
Twilight Princess was a much-hyped game in its time - a hype level rivaled only by Ocarina of Time. It didn't hurt that Twilight Princess came out in late 2006, when the release of the Wii propelled Nintendo to a new height of popular and financial success. Accordingly, Twilight Princess got a big budget, a long development cycle, and a lot of hands stirring the pot. Sometimes Twilight Princess felt like a dev team trying to smoosh together dozens of disparate ideas. A lot of awesome ideas made their way into the game, making for some incredibly fun gameplay and some really cool moments. However, the story was a bit of a mess at times, and there were a handful of places that would've benefited from additional polish.
Since Twilight Princess's release at the end of 2006, we haven't seen anything quite like it. There have been Zeldas since then, in both two dimensions and three, but they've generally gone for a brighter, more cartoony style. (Breath of the Wild took the gameplay in an entirely different direction altogether.) I'm not necessarily begging for a much-belated Twilight Princess 2, but there's no denying that Twilight Princess makes for a unique experience.
Playing Twilight Princess is infinitely more enjoyable than doomscrolling through social media, playing a shoddy mobile game, watching lopsided football, or doing a lot of other things you might do in your free time. It's also a lot better than the vast majority of video games out there. Sadly, it's not quite the Greatest Of All Time, but at least you get to become a cowboy and herd some goats into the barn.

Woof.
Finally, Twilight Princess is a game where you can pick up adorable puppies and watch them gaze lovingly at you. That is what matters.