As a rule, deserts form around the tropics, and rainforests around the equator. As with every rule, here too there are quite some exceptions, though the general pattern is easy to spot on the world desert map:
The reason for this pattern is mainly the wind. Near the equator, air from storms rises very high and moves away from the equator, losing all its moisture by the time it reaches the tropics. When the air falls back down to earth here, it gets heated up even more due to atmospheric compressing. This is the process leading to dry, scourching deserts on the tropics. Other deserts such as those in Central Asia are formed because rain clouds can't reach them. This might be because they're behind a mountain wall shielding off all moisture, or simply because they're too far away from any sufficiently large body of water. Deserts can also appear close to sea (e.g. the Kalahari Desert), where ocean currents make cold waters well up by the coast. Because of the low temperature, little or no water evaporates, giving the same result as when there hadn't been a sea at all.
(Interestingly, because cold water can hold more gas, cold upwelling ocean currents are usually pretty oxygen rich: a perfect condition for sea life. So while the Kalahari Desert, and other deserts close to sea are almost devoid of life, the ocean next to it is usually teeming with it.)
So to answer your question: if Africa were to move south, the deserts would (roughly) remain on the same geographical longitudes, and there for would appear to move north. If the continent would venture so far south the current Sahara region would lie on the equator, there'd be rainforests where the Sahara is now, and a desert in the Congo basin. Crazy stuff huh? ;-).
More questions? Do not hesitate and ask!