I'm not sure how much of the ancient monuments are open in December, but must-sees are the Colosseum, the Forum, the Pantheon and the Capitoline Museum.
Also try to get to the Vatican and the Sistine Chapel - you might be able to catch the Pope's Christmas message if you're lucky.
Other famous landmarks include the Trevi Fountain, the Spanish Steps, Piazza Navona, Trajan's Forum and Column. If you're into shopping, Via del Corso has a lot of upmarket places.
I've been to Rome 4 or 5 times and it's hands down my favourite city in the world.