In short? Poorly written, insipid fiction that is mostly harmless.
To expand upon that a bit, the Twilight series is really not all that different from most popular fiction. Very few of those authors have any real writing talent and instead just cater to the lowest common denominator. And there really isn't anything wrong with that. Sometimes, you just want to read some mindless entertainment. Kind of like how summer blockbuster movies work. However, that doesn't change the fact that Twilight is not a well written book series.
The biggest problem I have with the series is that the main character, Bella, is an author analogue, a reader analogue and a Mary Sue.
For people not familiar with those terms, an author analogue is a character that is a fictionalized version of the author. That is to say the author wrote this character to stand in for her. That's poor writing and given the context of the novel, somewhat creepy as well. There isn't anything really wrong on a basic level of wanting to live out a fantasy, but it certainly doesn't make for a well written or interesting character.
A reader analogue is a character that is essentially replaceable with that of the reader. It allows the reader to take the place of the character and act out the fantasy as well. Again, nothing wrong with that, but nobody can tell me this makes for a good, well developed character either.
Being a Mary Sue is probably the worst offense. Essentially it's a character with few or no flaws and even the flaws they do have only serve to make the character more "badass" or "cool". This is the worst type of character in my opinion as having a flawed, identifiable character is key to good writing in my opinion. People are, by nature, imperfect. We may strive to do good things, but we will fail. We might be an asshole to a co-worker one day. Maybe we harbor latent racist or sexist sentiment. Maybe we're homophobic. Something of that nature. Without those flaws, we really aren't human. When writing human or human-like characters, it is important to include some of those everyday flaws in order to, well, humanize the character. Bella doesn't possess those traits in my opinion. She's too perfect. All the boys love her. She has supernatural beings FIGHTING over her affections, all without her really doing anything. That's simply not a character I can identify with or even like.
That all being said, the books are harmless. Mindless, saccharine reading designed for pre-teens and teens to enjoy without thinking too much about it. They are the popcorn flicks of books. And there isn't anything really wrong with that. Are these books my cup of tea? Not a chance. Am I going to judge (outside of some mild teasing) people who really like these books? Nope. It's fine that people like these books. Live and let live.
As an added benefit, there is a whole new generation of people who are geeks about something. Being a Twlight geek is still a geek. And geeks tend to get into other geeky things. When the younger Twilight fans grow older they will find other things to be geeky about. I really think we're heading for a Golden Age of geekdom, and that's actually a great thing, regardless of how we get there.