Go into your music library, and find the playlist you would like to reorder. Click on the pplaylist, and in the top right hand corner you will see "Edit", click it. All of your songs will now have a red circle with a white minus on the left hand side, and three grey bars stacked atop one another. Click and hold those three grey bars on an individual song to reorder it in the playlist.
Answer is based off of usage in iOS 6.0.1