There really is no way to neatly classify succulents. Many are spiny cacti with fleshy stems without recognizable foliage. Others are euphorbs (related to poinsettias) that may resemble cacti, or may instead have fragile leaves on fleshy stems. Aloes and agaves have bold rosettes of fleshy leaves that obscure their wide basal stems.
The most familiar succulents are small docile perennials, like the many varieties of jade plants and iceplants. Some are flowering perennials like begonias. A few are even grown as annuals, like busy Lizzie (impatiens).
So what do succulents have in common? Well, that is a good question that is open to interpretation. Most would agree that succulents have some sort of fleshy succulent parts for storing water through dry weather. These succulent parts are most often leaves or stems. Yet, yuccas, dracaenas and ponytail palm that are no more succulent than palms, are considered by many to be succulents like related aloes and agaves.
Many succulents are remarkably easy to propagate vegetatively (without seed). Most aloes, some agaves and many yuccas produce pups, which are basal shoots that can be separated as new plants. Agaves that do not produce pups while young will likely produce many pups after they bloom and begin to deteriorate. (Individual rosettes die after blooming.)
Despite the nasty spines that make them difficult to handle, cactus that produce multiple trunks can likewise be divided. Cactus can alternatively be propagated as large cuttings; but because they are so fleshy, should be left out for their cut ends to dry and ‘cauterize’ somewhat before rooting. Many euphorbs behave much like cacti, but are even more hazardous to handle because of their very caustic sap.
The majority of small succulents, like the many jade plants and iceplants, are notoriously easy to propagate by cuttings.