OK, I found the answer myself. This is from the site
www.bonuswhores.com, which I am in no way affiliated with. I hope they don't mind me copying and pasting:
What is a skin? The best way to imagine a skin is this: Say Party Poker owns a real brick and mortar casino. They hire Joe Empire to bring in players to their casino. Joe gets paid a percentage of what each player wagers. To mark the players as Joe Empire’s, he brings them in through a special door in the casino called Empire. All the games, dealers, and the like are the same once inside the casino, but the player is considered to be playing for Joe Empire.
It is perfectly allowable to have an account at both Party and Empire. It is nice to have accounts at both to take advantage of the bonus offers at both sites. The software does prevent you sitting at a table with yourself, so no worries about collusion there.
The multi-table tournaments at Empire are different from the multi-table tournaments at Party. So when you play in a multi-table tournament at Empire, you will be playing against Empire customers only.