Any girl extremely obsessed with any celebrity, t.v. show, movie, book, or other entertainment/media who is participating in fan actitivities such as message boards,
fanfiction, or conventions.
Has been used in the past as derogatory but has become less and less in certain fandoms. Fan girling frequently happens with large amounts of
squee.
The noun would be
fangirl or occasionally
fengirl, though the later is usually used as plural.
Her whole message board was fan girling over the new episode.