Wow.
Ok I'm just going to throw this out there.
I am a twenty-five year old male (though twenty four when I read the books, technically) and I actually really enjoyed them, some of my favorites truth be told.
one thing that hasn't been touched on here is the sense of world building, I love world building, as an example, my two other favorite series are Harry Potter and the Dresden Files, and I can assure you my knowledge of their respective universes is expansive and exhaustive. for me the most engrossing stories are the ones I can just absorb little details from, and twilight is positively full of them, the special abilities of some vampires, the way food choice has physical (eyes, strength) and mental (more trusting, rational) effects on them. their histories.
I enjoyed the plot to be certain, and I do think the series in general has gotten a really bad wrap because of the truly atrocious movies, but the real draw for me was just the world building and the fantasy romance take on the romeo and juliet story.
I should mention that I was never really fond of romeo and juliet, for reasons that I'll only go into if asked.
I notice a lot of talk of manipulation and abuse, and if that was your take, ok, but I approached it the same way I approach any fiction involving vampires or magic (or vampires and magic) it's fantasy. it is fantastic by design. if you go into a fantasy romance about two teenagers (one of whom happens to be a vampire) and don't expect it to be creepy if examined outside that context in the real world, then well you're kind of missing the whole point.
I'm a huge romantic. just a big hulking girly-man about this kind of thing, and I'll tell you that though the story is told from Bella's perspective, I identified with Edward. his thoroughly explored feelings for Bella are what kept me reading into the wee hours of the night.
when I came across Twilight, I had wanted for years to find a girl for whom I could feel as strongly as he does for her, and the series just latched onto that part of me and I couldn't put it down.
I've actually been wanting to go back and read them (which writing this post has inspired me to do) and see how I feel about them, now that I have found my Bella. Maybe I won't enjoy Twilight as much this time, or maybe, I'll simply appreciate Jen even more.