A songfic is a fic throughout which the lyrics of a song are interspersed. They are notoriously bad.
Many songfics have lyrics that have been used for countless other songfics (such as 'My Immortal' by Evanescence), putting the fics in danger of suffering ficverse fusion.
Possibly among the most aggravating are the ones that use a song and get the lyrics wrong, as well as those wherein the names of locations and people are replaced by the authors with names from the canon—or worse, their own stories. If the song in question is one an agent likes, bloodshed is likely to follow, or at least a lot of twitching, sobbing, and an indulgence in copious amounts of Bleeprin. If the song in question is one the assigned agent hates, the results are, oddly enough, very similar.
The phenomenon of songfic is most likely due to the large numbers of careless young girls that write fanfic—very rarely is a songfic written by a male author. For many adolescent girls, surface image, culture, and specific subculture are very important to their identity, and part of that complex is the music they listen to. Thus, they're more likely to have an emotional attachment to a specific song, music genre, or artist, and more likely to get a plotbunny inspired by some song they know. Such writers make the leap to fangirl status by actually including the copyrighted song in the resulting fics themselves.
Songfics likely exist for the same reason that many Mary Sues suddenly begin singing songs the author likes or knows. It follows a very pepper-jack cheese logic: If I like these two things, they must obviously go together.
When venturing into songifcs, Glopsnerch earmuffs are highly recommended.