Filk songs are actually an old science fiction creation. At science fiction conventions, you can almost always find a filk singer somewhere. Usually parodies of well-know songs, using changed words or using new words to an old tune, filk singing is basically just sci-fi folk songs. A very good example is the ever-popular twist to "Jingle Bells":
Jingle Bells
Batman smells
Robin laid an egg
The Batmobile lost its wheel
And Joker got away!
While simple it's wording, its definitely filk. Most filks tend to be based off of popular songs or well-known songs that the audience is more than likely going to be familiar with. Like any song, they portray emotions, tell a story and provide entertainment, such as Wierd Al Yankovic's parody song "Yoda" of The Kinks' song "Lola".
(However keep in mind if you consider Wierd Al mainstream, this opinion would not apply. I don't consider him mainstream myself but just extremely amusing!!)
A good starting point to learn more about filk songs is www.filk.com, which not only has great resources of filk songs but some good background (more than I've given here). If you are on any science fiction mailing lists, someone has posted a filk song sometime or another.
Filk songs should not be confused with song fics, which is something we'll get into next week!