Many plants reproduce by runners and offsets.  Runners (or stolons) are long stems sent out by the parent plant, with a new plant at the tip.

You can root the plantlets several ways.  The safest method is to fill several small pots with loose, moist soil and pin the plants to the surface with a hairpin or bent paper clip.  In three weeks or so the plantlet should be rooted; check to see by giving it a gentle tug.  If it resists, roots have developed and the runner can be severed.  Or you can cut off several plantlets, root them in water, and pot them.  Easier yet is to cut off the plantlets and pot them immediately into individual pots.

Offsets (or suckers) are somewhat more difficult to propagate successfully.  Many succulents, as well as most of the bromeliads, reproduce by this method.  Offsets develop at the base of the plant and look just like miniatures of the plants themselves.

Cut the offset from the parent plant with a sharp knife, getting as many roots with it as possible.  Be careful, however, not to take so many that the parent plant hasn't enough to live.  Be especially careful with cactus--they're usually reluctant to give up their offsets.  Put the offset into a small pot filled with loose, moist soil.  Even if you can't get many roots, the offset may still survive.  In the case of bromeliads, you can take as many roots as you want because the parent plant will die anyway soon after it blooms and produces suckers.