Binary Fission

The complex Eukaryotic cell divides using a process called Mitosis, which is a form of binary fission. Like it's name implies, binary fission is literally "splitting into two". Mitosis, and the details of cell division in general, is not covered here, but to summarize things let's just say mitosis involves dividing a nucleus containing genetic material and rebuilding new cells around each copy (using existing material).

Simpler Prokaryotic cells don't have a true nucleus, so the genetic material is usually just floating around unresticted save for the other organelles that may be in the way. Once the genetic material has divided, the cell wall closes in and separates into two daughter cells.

The animated GIF on the left is grossly simplified, and is not representative of any specific cell. I made it to help illustrate the process as a whole. It is up to you to study further and draw your own conclusions.