These small (1/16" - 1/8") creatures feed by sucking the sap from plants. In small numbers aphids aren’t a problem, but under favorable circumstances they can multiply rapidly and create large colonies. In such cases they remove so much sap from the growing shoots that they become stunted and misshapen. They may also transmit virus diseases. The various aphid species attack a huge variety of crops, but they are particularly common on the Brassicas.
Aphids have been called the mice of the insect world, because they multiply so quickly and provide food for so many creatures.
The best way to control Aphids is to have lots of insectory plants (the Daisy and Carrot families are particularly good sources of food) around to feed their predators. If their numbers get too great you can wash them off the plants with a strong jet (I mean strong) of water. In extreme cases you could use insecticidal soap or pyrethrum.
Image: Clemson University - USDA Cooperative Extension Slide Series, Bugwood.org