Question: I am trying to replace a 15A electrical outlet. The existing one is attached to 12 gauge copper wire. There is a ground connection, one set of wire on the bottom screws, two sets of wire going into the holes in the back. Obviously, the existing one can accept 12 gauge wire in the holes in the back.

The new one cannot accept 12 gauge wire in the back (I read that code doesn't allow this anymore), so am I not sure what I should do.

Answer: 
The best way is to use a pigtail. Use a large wire nut to join four 12 awg hot wires in the back of the electrical box. One of these four wires is the pig tail that will come out and join to the outlet at the screws. Do a similar arrangement for the four neutrals (one a pigtail) and bring out the single neutral to the outlet. 

This arrangement also makes it far easier to mount the outlet after the wires are attached.

Don't forget the GND wire attached to the green terminal on the outlet.
You can pick up a 20-amp duplex outlet with push-in terminals that will accept 12ga wire, that way it can be a drip-in replacement. 20 amp outlets have a t-shaped slot on the wider pin.

However, the pigtail method is just as valid. Really a question of which one is easier / cleaner / more convenient. Personally I would go the 20-amp duplex route if I had one handy (unlikely) or I was going to Home Depot for something else anyway. If I have a pocket full of the right size wire nuts I wouldn't bother.