How you go about making new plants from old.

Cuttings
Cuttings are normally a piece of the stem: one that is actively growing (softwood or tip cuttings); one that has stopped growing but is not quite mature (half-ripe or semi-mature cuttings); or one that is quite mature and firm (a hardwood cutting: it may be dormant and have lost its leaves).

Take cuttings just below a node in the stem. (Nodes are where a leaf arises.) New roots are produced from below the node. Remove all leaves that will be buried and cut the remaining leaves in half to reduce moisture loss. Always take more cuttings than you need to allow for some failures. Always take the cuttings from young vigorous healthy plants for the best chance of success. The length of time rooting takes varies, but you can speed it along by dipping the cut ends into fresh rooting hormone.

Tip: Rooting hormone loses its potency quickly and an old or opened packet is useless.

Hardwood cuttings can be struck in the ground in the same sort of bed in which you would grow seeds, but soft and half-ripe cuttings need some protection. The simplest way is to pot them up in very sandy soil (3:1 sharp sand to regular potting mix), and enclose each pot in a plastic bag to keep moist and humid.

Grafting
Grafting is the union of one plant (the scion) with the roots of another (the stock or understock). It is a skilful process and most home gardeners will not need to master it. It is used to control the growth of the scion, either by giving it the benefit of more vigorous roots than it would make for itself (as when roses are budded on wild rose roots) or, conversely, by using a less vigorous, 'dwarfing' stock, a technique used especially with apples and pears to create smaller, more manageable trees.

To graft
Make a wound on the stock plant and insert into it a piece of scion (a cutting). Hopefully both stock and scion will callus together and grow as one. To do this, you must match the green sections (cambium) below the bark of both stems together perfectly. Use razor-sharp blades and a steady hand to make the cut. Disinfect the blades after each cut to minimise infection.

First cut a bud and sliver of bark from the plant to be grafted. Don't let it dry out.

Make a shallow cut in the bark of the understock. Lift the flaps and slip the bud behind the flaps. If the bark is stiff, water it well, wait a day or so and try again.

Bind the flaps together with raffia or budding tape.

Budding: The simplest form of grafting is budding (or shield grafting). Around midsummer lift a flap of bark on the understock and slip in a growth bud from the scion, trimmed so it has just a sliver of bark to support it. Cambium matching is automatic, and you bind everything together with raffia or plastic tape. Let the stock grow, cutting it off just above the bud the following winter. Come spring, the bud will grow away to start the branches of a new plant on the more desirable roots.

Cleft grafting: This is another option. Cut off the stock first, cleave it with a sharp knife and insert a scion in the cleft (or two, one on either side), matching the cambiums exactly. All the wounded surfaces are then covered with grafting wax and the graft is enclosed within a plastic bag to keep the scion moist. The best time for this process is towards the end of dormancy in early spring.

Source:Gardening: A Commonsense Guide (Murdoch Books)