The shoes you wear should be suited for the surface on which you play. Grass, clay, and hard court surfaces are different, and you need different kinds of shoes for each of these surfaces.
If you are an intermediate or advanced player playing on different surfaces, then you can choose multi-court shoes that suit all surfaces. Otherwise you'll need to choose shoes meant for a specific surface.
For a hard court, the shoes need to have ample cushioning to absorb the shock from the surface.
A tough upper is another feature in such shoes. The design of clay court shoes would be such that it prevents the clay from entering your shoes. The tighter knit upper gives stability as you move around.
For the grass court, the outer sole needs to have cleats or nubs that offer a better grip. The shoes you use on grass cannot be used on other surfaces. This surface calls for shoes that help you move fast.