There are many methods of propagation, but I'm only going to mention those methods either used on my own farm, or commonly used in my region.
Seeds......... I think everyone understands what a seed is. Seeds are found inside the "fruit" produce by the plant. The fruit can be moist, such as an apple or melon, or dry like the husk of corn or wheat. We eat many seeds, included among them : corn, peas, beans, cowpeas, grains, pumpkin seed, coriander, amaranth. Other seeds are consumed along with the fruit : cucumber, tomato, pepper, eggplant, tomatillo, ground cherry, strawberries, summer squash, pipinola, gourds, guava. Of course there are times when we discard the seeds, such as with cantaloupes, watermelon, pumpkins, winter squash, and most tree fruits. Seeds may be the most common way to start veggie plants for a garden.
Cuttings....... This is a method actively used on my farm. I'm constantly increasing the square footage of sweet potatoes, Okinawan spinach, cholesterol spinach, chaya, all started from cuttings. I occasionally propagate kale this way, but it's not as successful as the others. A cutting is a piece of the stem of the mother plant, often pre-rooted in water or directly planted into soil. Some plants require the use of a rooting hormone, but I don't bother with that. If it can't root on its own, then I simply don't propagate it this way.
Runners/Stolons...... Outside of my pasture grasses, the only thing in the gardens where I routinely use runners to make new plants is mint and strawberries. Mint may send out short runners just beneath the soil surface, or root along the stems that lay along the ground. Strawberries send out runners with the plantlets formed along the runner at the nodes. By the way, gardeners commonly call them strawberry runners since they run along the soil surface. Botanists call them stolons. Some taro varieties also will produce stolons that can be used for propagation.
Suckers ........are shoots that the plant produces, usually along the stem but sometimes from the stem at soil level or below. I use suckers to grow more pineapples and sugar cane. Although cane is commonly grown from cuttings, I like using developing stem buds (locals call these suckers) because they do better for me. I'm not sure if bananas keikis are termed suckers, but I looks to me that they are.
Root division...... This is another way to propagate sugar cane. Just take a sharp shovel and slice a chunk of the root out. Comfrey is propagated from root pieces.
Layering...... I don't use air layering, but I often use simple soil layering. To get more baby plants for stick oregano, I will pull down a stem to the soil surface so that part of it about 6"-12" from a stem tip comes in contact with the soil. Plop a rock atop it, then go back after a couple of months and harvest the new baby plant. I've also rooted mulberry and guava this way.
Crowns.... Pineapples and taro are routinely propagated this way. To get a pineapple crown, simply grab the spiky green top and twist it around until it pops off the fruit. To get a taro crown, slice it off the top of the corm, taking about 1" of corm with it.
Slips..... This is a term used to describe the plantlets that pineapples produce from the peduncle (region at the base of the pineapple fruit). I routinely use them to start new plants.
Plantlets.... I suppose that pineapple slips could be termed plantlets. At one time I had a type of walking onion that produced plantlets atop a central stem. A plantlets is a small developing plant produced right on the mother plant. When it drops to the ground, it roots and becomes independent of he mother plant. Taro produces plantlets off its corm that can be separated from the mother plant and planted.
Corms..... A corn is a swollen stem area at the base of the plant. The corm acts as the plant's storage unit. Taro has a corm. I have used the central corm instead of just the crown (huli) when I'm trying to produce as many new plants as possible, especially with a variety new to me. You can plant the corm on its side, or cut it into longitudinal quarters and planting them sideways. New shoots will emerge from dormant buds in the corm.
Bulbs.... Bulbs are underground plant storage units which represents the entire plant. An onion is a bulb. Examine an onion closely and you can identify a basal plate with root buds, basal nodes for bulblets, a central core that will produce the plant and flower, layers that represent leaves. Pretty nifty.
Rhizomes..... A rhizome is a continuous underground thickened stem (or along the soil surface, as is the case for some gingers) that produces roots and aerial shoots at the nodes. In grow ginger and turmeric, which are rhizomes.
Tubers.... A tuber is an underground storage unit along an underground stem or modified root. The first one to come to mind is the potato. It's a classic. A potato forms at the end of the underground stem. It grows new plantlets from specific points on the tuber, points called eyes. The sweet potato is also a tuber, but termed a tuberous root, as it develops along a modified root.
Grafting..... With grafting you take a stem or bud from one plant and match it up with the stem/root system of another in such a fashion that the piece is adopted by the root system. Thus you are taking the top of one plant and grafting it to the bottom of another. I've grafted avocados because it's fairly easy. I've not tried others, but I do have grafted citrus, peach, mango, and macnut trees. Why graft? It assures tha variety you desire since these trees grown from seed would not be true to the parent type. Grafting also results in the grafted tree bearing fruit earlier than one grown from seed. Fruit trees are often grafted in root stock that results in a dwarf tree, popular among home gardeners and old folk like me.