A boss song is a song that usually requires some form of Unlock, or is hidden unless a requirement is met (Eg. AA on your last stage, Entering an Unlock code etc). More often than not, they are harder than most of the regular songs, if not the hardest song for the current mix.
Boss songs are usually only seen after achieving a AA on the final stage, enabling the Extra stage, which is played in reverse, no recover and with another modifier (ranging from Dark to 1.5X.) Depending upon the game, beating this song One More Extra Stage/Encore Extra Stage is opened which is a tougher challenge since you fail when you miss one step (this includes getting a good.).
On home editions the boss songs are usually unlocked by beating them or beating a certain amount of songs, or through a Master or a Circuit mode. In some games which feature challenges (such as DanceDanceRevolution Hottest Party 3), a player may unlock a song if he/she passes it. And if the song in question is a boss song, it is also unlocked, no strings attached.
Starting with the arcade version of DDR X2, in addition to the Extra Stage and Encore Extra Stage only songs, a new category is made available where the player can choose a song from a set of 6. When medals from all songs have been earned, the player is immediately forced to play a new song as the Encore Extra Stage with only one life.
The Boss songs on In the Groove are generally unlocked by entering a code on opening select screen (After a coin has been entered and the select button has been pressed once.) They are the 13 footers of the games, and once unlocked are free to play by all.