Most people buy plants to get something dependable that will grow and flower the way it's supposed to, but others try for something different, unexpected and, possibly, better, by breeding their own. This means either picking out the best individuals in a group and letting only those set seed or painstakingly placing pollen onto the pistil (the seed-producing part of the flower).
One of the most common reasons for breeding your own varieties is to have flowers of a unique color or form. Perhaps you'd like pale pink California poppies. If you grow a variety that includes pink among the other colors, you can remove all but the plants with pink flowers, so that they will cross with each other. Do this over and over and you'll eventually get seedlings of the color you prefer.
Or you'd like daylilies with narrow petals. Or exceptionally large rhododendron flowers. You set the standard and you choose the parents that you hope will give you the result you want.
For your large rhody blooms, you may cross a tropical species with exceptionally large flowers with a hardier but smaller flowered variety, hoping to get a shrub with huge blossoms that will also take the low temperatures in your area, a new feature, a new type of plant that no one else has in their garden.
Many plants are grown for their ornamental leaves, for the shape and size or the patterns and color, or even the autumn foliage hue. You might grow a group of hostas from seed and choose the one with the most striking leaves to grow and divide, perhaps even selling it to other gardeners.
You might also make specific crosses between varieties, hoping to get a particular color or pattern. Sometime a shrub will sprout a branch with an unusual pattern, perhaps with yellow or white splotches. You can take cuttings from that branch and grow shrubs with completely splotched foliage.
Many edible fruit varieties are the product of crosses between different species or varieties of that fruit. Hybridizers are looking for storage quality, early or late ripening, and, of course, delicious taste.
Disease resistance is also a desirable trait in fruit production. If the leaves of peach trees in your area are often attacked by a particular disease, for example, you could grow 30 small peach trees from seed and select the ones whose leaves are disease-free.
Seedlings often vary in resistance to frost damage and by growing your own perennials or shrubs from seed, you can select the ones that do best in your coldest weather. Or you could cross a somewhat tender rose with large flowers with another, hardier rose whose flowers are less desirable to get a tough, hardy, large flowered rose, perhaps unique to your garden.