They indicate whether the song will be played or burned.
For example, if you wanted to burn a playlist to disc, but didn't want Track 7 on the disc for whatever reason, simply uncheck that song's box. iTunes won't burn that track. Same for playback.
To check or uncheck a lot of songs, hold down Shift to select more than one song (use the Apple key if the songs aren't in order), then CTRL (right-click) click one of the highlighted songs and click "Uncheck Selection".