Plants are increased by two basic methods: seeds or vegetative reproduction.
Vegetative propagation (propagation is the word usually used by gardeners to describe the process of creating new plants) includes several different methods by which parts of living plants other than seed are used to produce new plants. Plants propagated by vegetative means are duplicates of, and have exactly the same characteristics as, the parent plant. Seedlings, which are completely new plants, may or may not be like the parent; plants (called hybrids) from seeds resulting from a cross between two different plants may resemble either or neither parent or inherit one or more characteristics from both parents.
Vegetative methods of reproduction include cuttings leaf, stem, or root; division of a large plant, or of a tuber, corm, rhizome, bulb, or roots; and layering. Budding and grafting are other methods of vegetative propagation of new plants from old, but they are a bit tricky and usually practiced only by experienced gardeners. There is nothing difficult or mysterious about any of this, and knowing how and why it is better to use one propagating method instead of another for each kind of plant makes the whole thing fun rather than just another task.