Cacti have thick, fleshy stems outfitted with nasty spines instead of leaves. Agaves and related aloes have stout, fibrous stems that are mostly obscured by thick, fleshy leaves. (Only a few somewhat rare aloes develop bare trunks and stems.) What they have in common is that they all are succulent plants, collectively known as succulents.
There are all sorts of other succulents. Humongous saguaro cactus have hefty trunks and limbs. Diminutive impatiens (like busy Lizzie) are grown as annuals for their colorful and very abundant flowers. Many succulents have succulent stems. Many have succulent leaves. Some, like the common jade plant, have both succulent leaves and stems. Trailing ice plants, leafy begonias, and sculptural euphorbs (related to poinsettias) are all succulents.
Many succulents store water in their succulent parts because they live in dry climates. Because moisture is such a commodity where they live, cacti protect their succulent stems with sharp spines. Agave protect their leaves with sharp teeth. Euphorbs are equipped with caustic sap, and many also have spines like cactus have. Fortunately, most succulents are not so unfriendly.
Almost all succulents are remarkably easy to propagate from cuttings or by division. In the wild, pieces of prickly pear cactus that fall onto the ground will begin to develop roots through the rainy weather of autumn and winter, and be ready to grow into new plants by spring. In the home garden, cactus cuttings should be left out for a week or more so that the cut ends will ‘cauterize’ (Actually, they just dry out a bit.) and be less susceptible to rot once they get plugged into the ground or pots to grow roots. Alternatively, clumping cactus that develop multiple main stems from the base can be divided, although the spines make handling them difficult.
Most aloes and some agaves produce basal shoots known as pups, that can be split from the main plants to grow into new separate plants. Agaves that do not produce pups while young typically start to produce pups after a few (or many) years, as they mature enough to bloom. Many of the larger types produce an abundance of pups after bloom, since the main shoot dies as flowers deteriorate. If desired, one or more of the pups can be left in place or planted back to replace the parent plant.
Most other succulents are even easier to propagate. Small cuttings can be plugged wherever new plants are desired. Some can even be grown from leaf cuttings!