When you look "contemporary" up, it says: existing, occuring, or living at the same time, belonging to the same time. Which makes sense since contemporary books is set in our world - nothing supernatural or anything. You could call it realistic fiction if you like.
Another popular definition is: fiction that is set in the modern world without the slightest supernatural factor. Everything is real, nothing is made up.
Goodreads has this great page where you can choose a genre and then you can see new releases, giveaways, lists, popular books and more all tagged with the genre you've chosen. For contemporary click here.
Some of my favorite contemporary books are:
Fangirl by Rainbow Rowell (review)| Attachments by Rainbow Rowell | The Fault in Our Stars by John Green | Dash & Lily's Book of Dares by Rachel Cohn and David Levithan
Which genre should I tackle next?