Whenever you select a playlist, iTunes will display some key information at the bottom of the screen: how many songs are in the list, how long it takes to play the entire list straight through, and (in case you care) how many bytes of data it represents. (This last bit of info is most useful when you want to burn the songs in a playlist to a CD.)
Playing the playlist is much like playing a CD. Choose a song and click Play (or just double-click the song), and the rest of the songs follow in the order in which they appear in the playlist.
Sometimes you want to be surprised by a song. Particularly with large playlists (or your library), you just want iTunes to choose a song for you and play it. The Shuffle button at the bottom of the Source column handles this for you.
But what if you don't like random playing, and you don't like the order in which you've dropped songs into your playlist? It's simple to change the order of your playlist in a more permanent way.