Fangirls are typically senseless followers of a certain pairing, author, or story. A fangirl lavishes praise upon her subject of choice and is eager to defend its honour, should it be attacked.
Fangirls tend to crash sporking parties with their insistences that the author/pairing/story is the best thing since sliced bread. Fangirls are usually looked on with contempt and are often referred to for a cheap laugh.
This term is also commonly used as a verb: when one "fangirls," she (or he, possibly) hovers about a particular story, etc. in a fit of excited hysteria. Also practised by regular readers who enjoy mocking said fangirls.