Dead heading is the term we use for removing spent blooms from annuals, perennials, and even some shrubs or bushes.
Dead heading is important for a number of reasons:
1) For annuals and reblooming perennials, removing dead flowers allows the plant to set new buds and bloom again. Leaving the dead blooms on the plant will impede its ability to rebloom.
2) Even for plants that do not rebloom, removing dead buds helps keep the plants healthy. Especially during rainy seasons, dead buds and dead leaves hold the moisture close to the plant and can cause rot, mildew and insect infestation. In dry weather, dead leaves will generally dry and fall of on their own, but in wet weather, go ahead and remove any dead material from the plant before it causes trouble.
3) Removing dead blooms just makes the plant look better!
Dead heading is really pretty simple: remove the spent bloom and the attached stem down the the next leaf node below the flower head. You can take the flower and leave a couple inches of stem, but it will die off down to the next leaf node anyway, so do yourself a favor and save a step - it'll look better and the plant will be healthier. (Note: leaf nodes are the areas where one or more leaves grow off of the stem. Cutting of the stems can be done with scissors, clippers, and even your fingernails.
Roses are a little different and a little tricky when trying it for the first time. For single stemmed roses, the dead blooms and the stems they are on should be taken back to the second set of five leaves below the bloom. Depending on the type of rose, there may be up to four sets of three leaves under the bloom before you reach that second bunch of five leaves, so take your time and do a little counting. Remove the dead bloom and take off the first bunch of five leaves under it, as well as any number of groups of three leaves, ending where the stem meets the second bunch of five leaves. You may actually see some little buds starting in this area, so be sure not to cut those off! For shrub form roses that have bundles of blooms, remove dead blooms right where they meet the stem until the entire bundle is spent, then follow the same rule as with single stems.