Propagation by cutting is usually explained as one method. It is a family of methods, and the differences between them are the differences in material: what part of the plant you cut, how mature it is, and whether it carries a bud of its own.
Stem cuttings, in four stages of ripeness
Tip cuttings use the growing point of a herbaceous plant, complete with the terminal bud. Of all the stem methods this one roots fastest and needs the least time in the frame, because the bud at the top is already producing the hormones that push roots.
Softwood or tender shoot cuttings come from fresh growth that has not begun to firm. They are easy to take and they root readily, but they have no reserves and thin, soft tissue loses water quickly, which is why they fail when the air around them is dry. Humidity is the entire skill in this category.
Semi-ripe cuttings are taken from current-season growth that has begun to mature and change colour at the base. They carry enough stored material to survive while a callus forms, which makes them the most forgiving of the three and the reason so many shrubs are struck from them in late summer.
Hardwood cuttings are fully mature, ripened stems of woody plants, cut roughly 15 to 30 cm and taken in winter or early spring while the plant is dormant. They root slowly and need no coddling while they do, which is why they are the commercial way to make large numbers of plants cheaply.
Leaf cuttings
Some plants will make a new shoot from a leaf. A whole-leaf cutting takes the entire leaf with its stalk; a leaf-segment cutting cuts the blade into pieces, each of which can throw up a bud at the point where a vein is wounded, which is how one leaf becomes several plants. The leaf-bud cutting is the reliable version: the leaf is cut with its stalk and a section of stem so that the axillary bud travels with it, and that bud, not the leaf blade, is what grows into the new plant. Use it for species whose leaves alone root badly or root far too slowly to be worth the pot space.
For a plant where this is the normal method rather than the clever one, see how to propagate a monstera from a cutting, where the presence of that node is the whole decision.
Root cuttings
A root cutting is a length of root laid or stood in the medium, and the new shoots arise from it rather than from any stem tissue. The requirement is that the piece must be able to make adventitious buds, which rules it out for many plants and makes it the natural choice for the shrubs and perennials that root from pieces left behind when you dig.
Scales and bulbs
Scale cuttings apply the leaf-cutting idea to a plant whose leaves are scales wrapped around a buried stem. The parent bulb is separated, the individual scales are taken off with a sliver of basal plate attached, and each one raises a small bulblet at its base. It is slow, it is the way to multiply something that will not make offsets on its own, and the new plant is a proper bulb rather than a rooted leaf.
Bud cuttings
The last category is the smallest piece of all: a single bud with a little stem attached, taken from the leaf axil or from the crown of the plant. It is used when the material is precious, because one shoot contains several buds and each is potentially a plant. Because there is no leaf to feed the new roots, the piece has to be kept humid and warm until the bud breaks.
Everything above is a variation on the same mechanism, explained without the plant names in what a cutting actually is. For the season-by-season version, including which shrubs are worth taking while the wood is half-ripe, see the June cutting list, and for the dormant hardwood treatment in full, with soaking and heeling-in, the dogwood hardwood protocol.