Okay, so there a trillion and one Harry Potter ships out there, but I'll bring you a little info on the major ones, involving Hermione.

The Good Ship - Side Kickers

The Sidekickers. Ron and Hermione shippers ship this particular ship because of the constant bickering and aruging between the two characters. This ship is also part of a bigger sub-category of OBHWF (One Big Happy Weasley Family) and relates with Harry/Ginny and Neville/Luna.

Harmony - Pumpkin Pie

The hero/heroine ship. This ship and the ship above have constant wars as to who Hermione will end up. The whole basis of this ship is that Harry and Hermione are kindred spirits - they are always together and help each other out. This relationship is of mutual understanding and respect. This relationship relates with Ron/Luna and Neville/Ginny.

Dramione - The Leather Library

The whole notion of love/hate. There isn't much canon proof, but there sure are some talented people out there who can make you think otherwise! This ship is one of the funniest and has the whole "house unity" and "I hate you, I love you" sense to it. A very popular ship.