Unicellular organisms reproduce asexually in a process called replication. Unlike with multicellular organisms, reproduction and cell replication for unicelluar organisms is the same process.
Most unicellular organisms reproduce by replicating their parts and then dividing in two. This process creates more than two new cells; it creates two new organisms.
Theoretically, this form of asexual reproduction results in two identical cells. In the case of organisms such as bacteria, however, the DNA has a fairly high mutation rate. This rapid mutation rate is why bacteria are capable of evolving resistance to antibiotics.
Replication is also called binary fission and is the most common type of cell division.