Vegetative Propagation Vs. Asexual Seed Formation
Asexual reproduction in plants is most often carried out through vegetative propagation---that is, a new plant arises from, say, a leaf instead of a fertilized seed. Another type of asexual reproduction does produce seeds, though without the fertilization that happens between the ovule and sperm of sexually reproducing plants. There are three kinds of asexual seed reproduction and several ways that plants carry out vegetative propagation. Asexual reproduction almost always produces clones of the parent plant.
Some plants grow underground stems called rhizomes. At intervals along this stem, roots grow and shoots of a new plant appear above ground. Grasses possess rhizomes. Some plants, like the Irish potato, make tubers, which are simply modified rhizomes. The tubers produce buds (eyes) that can become new plants.
Some plants produce new plant shoots from their root systems. This form of vegetative reproduction, which is used by apple trees, among others, is called suckering, and new shoots are called suckers. You can see suckers growing sometimes at the bases of apple trees.
Plants like the strawberry produce stems that grow along the ground, creating new plants at intervals along the runner, complete with roots and leaves. The runners are also called stolons. Strawberries, incidentally, can also sexually reproduce.
Plantlets are new, small plants that grow on the leaf edges of a parent plant. Eventually the parent and offspring plants become separate. Duckweed employs this reproductive strategy.
Bulbs and Corms
Bulbs are actually stems surrounded by leaves, which feed the stem during growing season. The layers of an onion, for instance, are actually these leaves. Lilies are another example of plants that put out bulbs for future plants. Corms are similar to bulbs, except there are no leaves, only a swollen stem that feeds new growth. Gladiolus make corms.
Growth from Damage
When a plant is damaged, sometimes new growth emerges from the old. Tree stumps often produce new shoots; broken leaves can also take root.
Adventitious embryony, diplospory and apospory are all terms referring to the ability of some plants to produce fertile seeds without the mother cells in the flower being fertilized. Seeds might grow from a plant's female cells or cells that are not sexual. Plants growing from these seeds, then, have mothers only. This way of propagation is increasingly called apomixis, a term that is sometimes still used to describe any form of asexual reproduction. Some squash have this reproductive ability, useful in the absence of pollinators like bees.