To "deadhead" plants means to remove their spent flowers. For many plants, deadheading promotes more flowering on the plants for that year than would occur without such plant care. For soft plants, you can deadhead by hand; other times, you'll need to deadhead with scissors or pruners.
It's a good idea to deadhead your annual plants (and many perennial flowers, too) if you wish to achieve maximal landscaping with four-season interest. Deadheading gives them the chance to continue blooming throughout the summer and, in many cases, into the autumn.
Without deadheading, some annual flowers will "peter out" prematurely, robbing the landscape of the color they can provide in late summer / early autumn. And the comparatively short blooming season of some perennials can be extended if you deadhead them.
"Deadhead" and "pinch" are very similar terms. Some people use them interchangeably. Others make a technical distinction, insisting that you "pinch back" a plant before it flowers, to make its vegetation bushier; whereas you can only "deadhead" a plant, by definition, after it has flowered (since it's the flower "heads" you're removing).
Even if you draw such a distinction between the 2 words, deadheading and pinching back do operate under a similar principle: they're both all about channeling a plant's energy into a direction you find more agreeable than the "natural" direction. When you deadhead flowers, you are channeling energy away from seed production and into further flower production. In essence, to "deadhead" a plant is to trick it into forming additional flowers, in its attempt to (finally) produce the seed it set out to produce in the first place, before you deadheaded it!
Such botanical reasons aside, people also deadhead plants simply to keep ugly, shrivelled-up flowers from marring the appearance of a planting bed.