Plants such as aloe vera, bromeliads and African violets may produce small plants right next to the original plant. These little plants are called offsets. Using a sharp knife just cut them away from the parent plant, making sure the offsets have roots. Then re-pot immediately.
Miniature plants, called plantlets, can be found at the end of long stems called runners. The strawberry begonia and spider plant will propagate naturally in this way. Other plants, such as piggyback plant will form miniatures on their leaf surface. Plantlets should not be removed from the parent plant until they have roots. To help this occur, pin them down to soil with a paper clip, perhaps in a second pot, and wait for the roots to form. This may take 2-3 weeks. After roots form, snip and remove each plantlet from the parent plant and re-pot.
Spider plants root easily. Often their plantlets form short white aerial roots while still attached to the parent plant. Many gardeners will remove these plantlets before a complete root system is developed, and re-pot them successfully.