One example of asexual reproduction is something so simple, called binary fission. It occurs in the simplest of cells known as the Prokaryotes. These include bacteria.
Bacteria usually proliferate, or reproduce, at exponential rates. They primarily do this using binary fission. Here, the parent cell simply divides 'in two' and forms two genetically identical daughter cells.
In Eukaryotes, which are higher organisms on the evolutionary scale (and more recent), asexual reproduction is called mitosis.
Some examples of mitosis are:
Plant cells in a tree root dividing as the root lengthens;
Your skin cells divide asexually. They also use mitosis. Your skin is therefore 'shedding' all the time (has a high turnover rate).
Lizards' tails. These reptiles are also called anoles. In Florida, these lizards often lose part of their tail when a predator grabs at it. The tails can grow back; they do so through mitosis.