A 'gripper' is a component of the robot used to manipulate an object loose from the robot itself. This can be a ball it needs to pick up, or the dirty socks it was programmed to find and dispose of. Grippers exist in many varieties ranging from simple; a thong-like design to complex arms.
How complex your robot's gripper has to be depends on what it's supposed to do with it. Picking up a ball requires a different approach than picking up socks.
Kind of grippers
Grippers are commonly modeled after tools, machines or the human hand. The choice behind a gripper doesn't differ much from the choice of the tool on a excavating machine, forklift, crane or other machine.
Scoops are very simple 'grippers'. These can be flat, flat with raised edges or more complicated shapes. These 'grippers' are easily to control as they can be build with minimal moving parts: being able to move up and down is sufficient for many purposes. Forward Tilting is common too and may be necessary to 'unload' the scoop.
Tongs are more complicated than scoops but allow better control over what is picked up. For example: with a scoop it's very hard to pick up a glass and put it down somewhere else. With a tong-like gripper this is trivial. Otherwise is it much harder to move a ball with a tong than with a scoop.
Tongs differ in the number of fingers and their shape. Most tongs have 2 fingers, one of those could be fixed.
Many designs for robot hands have been tried including quite anthropomorphic hands.
Robotics has also led to a renewed interest in the 'design' of the human hand. The asymmetry of the human hand and in particular the fact that a rod like object gripped in the hand naturally falls at about to the axis of the hand gives much greater flexibility in tool use than a symmetrical design would.
Degrees of Freedom
Grippers also differ in their degree of freedom. Basically there are 6 degrees of freedom: 3 translations and 3 rotations. See Theoretical Mechanics about these degrees of freedom.
But generally, unless you're building an arm, you wouldn't need all 6 degrees of freedom (DOF). You get one DOF for free on a mobile robot: it can already move forwards and backwards on its own. Although adding gripper movement along this direction will increase your robot's accuracy, it's not really necessary.
How many DOF you need depends on how complicated the movement of your robot has to be. If it only needs to pick up a glass and move towards a goal and put it down again, it would be sufficient to make a gripper that can tilt back (towards the robot).
If you intend that your robot will have to be able to pick up Lego blocks and build a structure with them, you'll need more DOF.