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In Game Center, change the account in the "Me" page to the account you want to copy from. Note: You must know their password.
Run Clash of Clans. A tab should show up saying "(Player's Name) with (Town Hall Level) is going to be copied onto this Village" Touch yes to copy, touch no to go to your village.
Note: If you keep the other person's account on Game Center, every time you run Clash of Clans it will ask you this.
Copying someone else's village will erase the village that was already on there, so make sure you can save it before pressing yes.