Admission #1: To the best of my knowledge, I have never heard a single song by The Velvet Underground. Until today.
Admission #2: When I checked out
The Velvet Underground and Nico from the EPFL, I really wanted to trash it. I hate VU followers like The Pixies, and I think Lou Reed's "Walk on the Wild Side" is one of the lamest rock songs ever.
Admission #3: I think this album is amazing
You don't need me to rehash what's been said a thousand times about
The Velvet Underground and Nico. Is it melodramatic and scarier-than-thou? Yes. Is it nihilistic and self-absorbed? Yes. Is it caught up in some ignorant and short-sighted viewpoints about life? Yes. Does it push boundaries? Absolutely. Do Lou Reed and Nico have annoying voices? Yes, but they're the right voices for the music. Is it difficult to listen to? At times, but in a good way. Is it unlike anything that came before it? No, but it changed everything that came after it.
Music: 
The album possesses a timelessness that eluded most bands from the late '60s. Sure, a few of the songs are dated: "Run Run Run" sounds like a Stones throwaway, and the backup vocals in "There She Goes Again" are generic trippy drivel. As a whole, though, this could've been recorded at any point in the past four decades. I understand why it's received so much praise over the years.
My favorite moments are when the band sinks into an almost abstract sonic assault, like on "European Son." They weren't relying on mounds of guitar effects, impenetrable layers of electronic samples, or overdeveloped technical prowess to create a dense wall of noise; they were simply taking what they had and wrecking shop with it. Hearing this song makes me realize that I was correct in dismissing
Boris as a bunch of wankers.
Packaging: 
The cover,
Banana by Andy Warhol, is pretty awesome, even if you're not a Warhol fan (which I'm not). It's simple, it's unique, it's eye-catching, it's immediately recognizable, and it's impossible to imitate. It has to be one of the greatest album covers in the history of rock. The inside of the CD jacket isn't nearly as impressive, though. Some of the photos are dark portraits of forboding characters staring at the camera, while others are dark portraits of forboding characters staring at the camera with cheesy lighting effects on their faces. It was probably cool at the time, but now it just looks corny.
Listen if you like: art rock, punk, indie, goth, noise, doom metal, experimental music, drug music, psych... really, I guess you should just listen if you like any rock music from the past 40 years.
If it were food, it'd be: A Twinkie. It's as fresh today as it was 40 years ago.