The most natural thing in the world for most plants in their first, actively growing stage to do is grow UP. Not branch, although we might want them to, not produce blooms or fruit or flowers or fantastic foliage, just grow straight up toward the light. If you've ever grown seedlings indoors, you'll remember how they stretch up toward the light source, even becoming "leggy." Many perennials grow so tall they flop over and need to be staked. Nature is anxious for plants to grow upwards and get energy from the sun.
A goal in life
A plant's goal in existing is to reproduce. Whether or not this characteristic has been bred out of your particular cultivar, originally, plants want to carry their genetic material on to the next generation. Maybe that means making lots of foliage so the tubers will be big for next year's plants, or making lots of seeds, or spreading by underground rhizomes. The beauty or bounty we get out of a plant is usually only secondary to its main reason for existence and sometimes completely unrelated.
How can I help?
While a plant is actively growing UP, it won't make branches, or side growth. In fact, its ability to branch (or make lateral growth) is suppressed. That's why you find climbing roses like mine which were 7 feet tall with blooms only at the top. (I don't know anything about roses, but I believe they should be trained horizontally rather than pruned—at least that's my next tactic with mine.) Once you pinch or prune the top, side shoots will develop, giving the whole plant a bushier appearance. When tomato plants have a certain number of leaves, they know it is time to start branching and making flowers (which of course turn into tomatoes).
You probably already pinch (with your fingers) your basil and coleus to stop them from blooming. This will work on most herbs and plants which you wish to keep for their foliage alone, like dill and thyme (although I love to see thyme in bloom). Pinching is also a good idea to encourage branching in sweet potato vine, so you don't get long ropes of SPV trailing from your containers. You also may know to pinch your chrysanthemums to encourage bushy growth, but the general caveat is no pinching chrysanthemums after July 4th. Some people pinch them twice, once on Memorial Day and once again on the Fourth of July.
With plants like iris, crocosmia, or foxglove, which bloom in a row up a stalk, pinching off the dead blooms will allow the ones yet to open to receive all the plant's energy.
Some perennials grow so tall that they flop over unless staked, which to my mind is a pain in the neck. If pruned before flower buds have formed, these plants may be induced to be shorter and bushier. I tried this with my balloonflower last year, which is so close to the path that I always end up running over dozens of flowers. Last year I tried pruning a few ... success! This year I had my daughter prune all 20—I can't wait. I can already see the bushy new growth. This should work on helenium or daisies, next I'm trying it with yarrow.
I recently read a fascinating article in favor of pruning tomatoes here and a counter-argument here. I had never thought of pruning tomatoes before; I'm always so happy that they're growing at all that the idea of pruning them seems cruel and unusual. I may try it if I ever get blasé about growing tomatoes, though.
And woody shrubs and trees, unfortunately, need serious pruning instead of just casual pinching as you're strolling through the garden. As Sally Miller says in this article, you wouldn't prune off your child's arm. But you do need to prune your flowering shrubs and trees, usually as soon as they are done flowering. I even printed out this terrific article by Paul Rodman about pruning trees to give my husband a big hint.
Unfortunately, the same principle that works in making your plants grow bushier sometimes works the same way with trees, shrubs and weeds. You've probably already experienced this. You think you've cut it in half and it comes back twice as big as before. Grrr! You may need to investigate painting the stumps with various substances or just cutting them over and over and over. Eventually, deprived of all energy from the sun, the stupid thing has got to succumb. Right?
In the case of flowering trees and shrubs, like forsythia, vitex, azaleas, and so on, we generally are pleased when they are bushier and have more flowering branches. With most flowering things, you need to find out whether they flower on old or new growth to know when it's safe to prune them, so you don't accidentally prune off buds that were about to flower. Pruning produces a healthier, bushier, happier plant. So now that you know what you need to know, it's easy enough to find out for your particular plants!
Picking is maybe the broadest category, because I am here including all types of deadheading, harvesting, and so on. When we deadhead annual flowers, the message gets sent that the plant has not completed its life's work (creating seeds), and instead of spending energy converting the dead flower into a viable seeds, the plant will instead refocus its energies on flower production. The same is only slightly less true for perennial flowers; if deadheaded regularly, we can often coax a second bloom put of them where none was expected.
Think of the cut-and-come-again type of zinnias. The more flowers you pick, the more you get. Lettuce and spinach often respond this way too, where the more leaves you harvest, the more will grow to replace them. Basically, you are postponing the moment at which the spinach or lettuce plant feels it has stored up enough of the sun's energy in its leaves to produce seeds (or "bolt"). If you ignore the spinach plant for a few weeks because you don't have time for a fresh spinach salad, those tender young leaves will have been replaced by an unappetizing seed stalk—because it thought it was time to make seeds and you didn't tell it otherwise. If you had kept plucking those leaves every day or two, whether or not you would have postponed the eventual day of seed stalk formation, at least you would have enjoyed some lovely spinach in the meantime! (Carrie, this means YOU.)
Flowers which are tiny and grow in mats, like creeping phlox, thyme, dianthus or impatiens, will often benefit from being sheared mid-season. I have often bought tired, leggy impatiens plants cheaply in the middle of the summer, divided them, cut one-third of the plant off, and repotted them, and been repaid with lush new growth!
Many flowers are not as precious or fragile as we might think. Pick them, bring them inside, and more will often grow to take their place. I'm not an expert on the zillions of species and cultivars out there, but take a chance! Your plant may surprise you. Now please excuse me while I start pruning; the spinach, I'm afraid, is a total loss, but I picked my lettuce today, and I've been pinching all along.