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.
Pups of the most familiar type are canine. Most are puppies of domestic dogs. Some are foxes or wolves. A few are seals, otters or various other animals. Garden variety pups do not get much consideration though. They develop into new progeny of plants rather than animals. Unlike animal pups, they are genetically identical clones of their single parents.
Pups of the horticultural sort generally develop from subterranean stems, such as corms, tubers or rhizomes. Most emerge from formerly dormant buds. Some grow adventitiously from roots. Pups appear as basal sideshoots against or close to their parent plants. They efficiently disperse their own roots to eventually survive as new and independent plants.
Most plants that produce pups are perennials. Some are so proficient with the technique that they grow into substantial colonies of genetically identical clones. For some, it is the primary method of replication. They actually prefer to replicate vegetatively than by seed. Some trees and woody plants do the same, but their clones lack the same classification.
For many perennials, propagation by division is merely the separation of pups from their parent plants. Most types of banana trees eventually generate a few surplus pups. Some types generate an abundance of pups to share with anyone who wants them. Separation of their pups is very similar to division of the smaller rhizomes of canna or lily of the Nile.
Individual rosettes of various species of Agave and a few terrestrial species of Yucca are monocarpic. In other words, they die after blooming only once. However, instead of dying completely, most generate several pups prior to bloom. Some produce pups excessively! Such pups can be difficult and hazardous to separate, but can replace their own parents.
Mediterranean fan palm is one of only a few palms here that generates pups. That is why it develops multiple trunks. Surplus pups can survive separation to grow as distinct trees. However, because of their densely fibrous roots, separation of intact pups can be difficult or impossible. Sago palms, which are not actually palms, are more compliant to division, even for rootless pups that develop on their stout trunks.