On an iOS device, if you have music synced with one computer, syncing with another will remove all the music from the device first.
So if more than one iOS device is involved thing appears getting more complicated.
To remove a playlist from an iOS device, go to the Music app, select Playlists, and swipe from right to left on the playlist. You'll get a delete button. In the iOS device's case, deleting a playlist will delete the songs from the device if the song isn't used in another playlist.
I tried the above on my iPad(third generation) and iPhone 5 and found the behavior is different as follows:
Music->Playlists-->open a playlist->Swipe left to right on a song then "Remove" is displayed. I assume that the song is removed if I tap on the "Remove". Note that nothing happened by swiping left to right on a playlist.
To remove that playlist in iTunes, right-click the playlist and choose delete. Songs will not be deleted because they'll still be in your iTunes library.
OK, I understand this.