The first thing to check is to make sure that your plant is a standard rosette African violet and not a trailing African violet. Most plants commonly sold are standard rosettes.

The little plants growing on the sides are called "suckers". Suckers will diminish the ability of your plant to grow well and flower well.

To remove the sucker from the plant you need to get a tool with a small sharp hook on the end, like a dental probe (those nasty hooked things your dentist uses), or a doctor’s disposable scalpel. These tools are called "sucker pluckers".

Take your sucker plucker and work it down behind the base of the sucker, where it is attached to the main stem of your plant. Carefully cut it off.

You can plant a sucker by putting it’s base into normal potting soil. Putting the new plant in a plastic bag to make a high humidity space will help the plant during it’s initial rooting period.