Generally you want to deadhead when the petals start to fade. A good example would be petunias - their flowers wilt, then the petals dry up and fall off. You can remove them when they wilt since they are no longer serving their purpose of looking pretty. The goal is to remove the old flowers before they turn into seeds, to encourage the plant to try again by producing more flowers.

Some flowers have interesting seed heads that you may want to leave on, or seeds that attract birds (echinacea and sunflowers come to mind). Some flowers won't bloom again even if you deadhead them (like irises). This article has some good basic info on deadheading perennials. For plants you can't find in a list, if you search for "[name of plant] deadheading" you can usually find something.

Generally, deadheading is good for annuals (plants that grow for one season and die when frost comes). Deadheading encourages continual flowering. When you pinch off faded flowers, be sure you pinch the seed pods that are attached under them, too. Just removing the petals doesn't do anything.

For perennials, you can leave the dried flower heads on for the birds to eat the seeds in winter. Some perennials look nice in the snow. Or...you can cut them down if you don't want to deal with reseeding or to lighten your spring chores.