A gaming mice connects via USB to your computer. You can connect it either to a desktop computer or a laptop. A gaming trackball acts in the same way as a mouse. It uses a ball and not a scrollwheel.
Both are used to control movement within video games. Some mice have extra buttons that can be used for special functions. You might press the button for a crosshair to be activated. Trackballs typically don't include any other buttons.
Most gaming mice work well with both right-handed and left-handed users. They are better suited to right-handers.
Right-handers should avoid trackballs because they aren't as precise.