Since the differences between the two have been explained well, I will focus on what they are both known to do. Androgenic and anabolic steroids exert effects on the body that are beneficial for athletic performance. Many users of these steroids also report the following effects: Increase in sexual drive, acne, increase in body hair, and aggressive behavior.
Testosterone has two main impacts on the body, and its androgenic impact is linked to male (from the Greek, "andro") sexual characteristics such as the way somebody speaks and the prevalence of body hair. The anabolic impct of testosterone is related to what we normally associate with steroids, which is muscle growth and the development of muscles.
Anabolic effects are the ones that have to do with cell growth. These are processes like the increased production of protein or development of muscles. Androgenic effects are the ones that have to do with masculine sex characteristics. These include things like increased growth of body hair or a deepening of the voice.
Anabolic effects increase protein synthesis, which is then translated to cellular to cellular tissue increase, a very beneficial effect when used medicinally in "wasting conditions." They also increases androgenic effects by increasing male secondary sex traits, like voice deepening etc.