Palm grip is the most prevalent way to hold a mouse for most gaming enthusiasts. A palm grip is comfortable and allows you to rest your fingers on the mouse buttons. This grip is suitable for most games, but some types of games require a different mouse grip.
Claw grip is similar to the palm grip, but you keep your pinky finger off the table and your thumb on top of the mouse. You can use this if you feel that you have less precise aim with a normal palm grip or if your hand gets tired easily. You can also use it to rest your palm on the table when you are not using the mouse buttons.
Fingertip grip is used by keeping your fingers off the buttons and resting them only on the very edge of the mouse. This grip provides good maneuverability, but prevents you from performing many actions with one hand – for example you cannot type while using this grip (I still recommend getting a keyboard wrist support though). I would recommend this grip mainly to people who play fps games where positioning is important.
Stick grip is just holding the bottom of an “ordinary” stick in between your index finger and thumb, without pressing any other keys down with another hand. If you are used to holding your forearm up or have long arms this is also an option.
Forearm grip is not suited for most games, but if you are playing a game where it is vital for positioning (e.g. Starcraft 2) and you use an ergonomic keyboard wrist support I would recommend using this grip as well. You can rest your forearm on the table without the support, but the support significantly increases accuracy over time with practice.
Closed fist grip is only suitable for players who have severe gaming mouse hand pains – resting your fingers on the buttons will decrease pain dramatically in most cases of carpal tunnel syndrome or similar issues. This is nearly never recommended by doctors though, therefore this information should be taken seriously.