Filk songs, in SCA usage (the term predates the SCA), are songs that are either old songs (either traditional, medieval, or modern songs) with new words, or just brand new songs, relating to SCA activities and lifestyle or amusing medieval content. They are often sung at Bardic Circles, while getting drunk around campfires, or on very very long car trips to camping events. Some songs are incredibly witty and amusing, others are so awful they make your head bleed, but either way, it's a fun activity for SCAdians to get involved in.
The word "filk" may be used as an adjective, the collective noun for filk songs, or as a transitive verb.
When filk lyrics are written down you may see the abbreviation "TTTO" near the title, which means "to the tune of..." followed by the title of a well-known song.
Not all new songs are filk, although to be honest the definition of what constitutes filk is fluid and fuzzy. A song written in a period style and with few or no OOP references is not neccessarily a filk song simply by virtue of having been writing in 2006 rather than 1066. It might just be a song. For a more detailed discussion, try http://en.wikipedia.org/wiki/Filk
Filk songs are also heard in amongst other communities and clubs, (eg. roleplaying, science-fiction/fantasy (where the term originated), choral), relating to their worlds and activities. The name was originally a typographical error for "folk song", and it was widely adopted by the science-fiction/fantasy community.
In many ways, filk songs serve the same function as traditional folk songs, in that they preserve lore and pass on cultural knowledge (such as it is) to newer SCA participants.