From The TimeSplitters Wiki

The Fists are your last resort for when you've run out of all other options. Surprisingly, the fists can be very powerful. You'll always have them with you no matter what. In the (exceedingly rare) case that you run out of ammo, fists become your primary and only weapon until ammo is picked up.

Summary

Quite simply, and obviously, the fists are standard melee attacks, with your chosen character unleashing a powerful punch, and will continue to do so as long as you hold the fire button (Or melee button in Future Perfect). Every character can use fists, and there is no difference in appearence, power, or speed between characters. The only possible advantage is height, in which case small characters such as the monkey, which is more nimble and to a degree faster, can be more efficient with the attack. At the same time, the defending character has an increased chance of getting a headshot, which basically nullifies the benefit.

Tips

The best place to aim for when using fists on an enemy is the back of the head. If you can get up behind them while they're fighting someone else, you can kill them before they even see you. However, if they do see you, you should keep attacking and moving randomly, but staying as close as you can. This will make you much harder to hit with most Weapons and you shouldn't have too much trouble evading.

If you wish to, it is possible to make a game with excusively fists. This is done by selecting a custom weapon set, and turning all slots to "none". This is fun when playing with friends.

Trivia

• The fists are visible on screen in TimeSplitters: Future Perfect unlike in TS1 and TS2.
• In TS2, the fists appear to be a open hand, rather then it being clenched.