Aphids suck on the juice of a plant through its leaves and stems. They then digest these plant juices and excrete a sticky, sugary 'honeydew' which may attract other garden insects like ants.
Small populations are generally not a major problem for gardeners,however, they are really fast at reproducing and large populations can take over a crop if not controlled. Many gardeners buy containers of live lady beetles (ladybugs) to spread over their garden in an attempt to curb population growth of aphids. Recognizing and allowing fire beetles and parasitic wasps to visit the infested plants can also aid in reducing aphid numbers and damage. Chemical pesticides that specifically kill aphids also exist, but will likely require multiple applications. Female aphids are mostly wingless, but winged ones will return to their original plant after starting populations elsewhere. They can be mechanically removes (killed by hand) but their small size and fast escapes make it a inefficient means of removing them.