I want to replace an older "round" outlet with a new square one. On the back of the old outlet there are two holes where I am used to seeing only one. So, there are two holes on the top left, two on the top right, two on the bottom left and two on the bottom right. I have three white and three black wires coming out of the wall. When I look into the receptacle in the wall, there are two sets (a black and a white) coming from the bottom of the receptacle and one set coming from the right hand side - none from the top.
I don't know how to connect all the wires to a normal outlet and I am unable to locate any new outlets that have this many holes in it. Help, please.

This link should give you exactly what you need.

Pigtail Connections - Make Pigtail Wire Connections (http://electrical.about.com/od/wiringcircuitry/ht/pigtailwireconn.htm)

Use of the quick connect holes is not recommended. They are famous for developing bad connection after a period.

One of the first things to look for when ask; "All of the outlets in my bedroom quit working. What is wrong?"

This link should give you exactly what you need.

Pigtail Connections - Make Pigtail Wire Connections (http://electrical.about.com/od/wiringcircuitry/ht/pigtailwireconn.htm)

Thanks TEV, I'm guessing I pigtail all three black together and all three white together, then run an additional wire from the pigtail grouping to the outlet?

You can easily see how this product line from Ideal industries IDEAL INDUSTRIES, INC. - Term-A-Nut Twist-on Pigtail Wire Connector (http://www.idealindustries.com/products/wire_termination/twist-on/term-a-nut_pigtail.jsp) addresses the problem perfectly.

They combine the wirenut, stranded wire and pigtail together and end in a Fork terminal. It makes the job quick and simple.

"The additional wire" is the pigtail. The pigtail should be at least 6" long.

"We (AMHD)" doesn't like poking wires in the back of the outlet especially the friction fit version of outlets.

Pigtailing the neutral is ALWAYS a good idea.

Mugs, when you bundle your 3 wires and add that extra wire that is the pigtail. You should only run a single black and a single white and green to a receptacle.

Thanks Everyone for all your help! I can take it from here.