There are multiple "views" in iTunes that allow you to play songs. As Tetsujin mentioned in his answer, iTunes only plays songs that are checked. However, this is only partially correct.
When you are viewing your songs from the Songs view, which is just one long list of all the songs in your library, then this is correct. Songs are played only if the checkbox is checked.
However, there is also an Album and an Artist view. To play an entire album, using the Album view (or Artist view) makes more sense, as you can select the play button for the album, and all the songs will play.