From The TimeSplitters Wiki

The Game Cartridges are mini-games dispersed throughout TS2's Story Mode. After you find one, you can play the game through your Temporal Uplink anytime by either pressing the Y button or Triangle.

Anaconda

Anaconda is found in Siberia (TS2) in any difficulty mode in the first room inside the building opposite the river on one of the shelves.

Anaconda is very similar to the Snake game found on a lot of cell phones. You start out with a small "snake" and as you run into more dots, your snake gets longer and it becomes harder to control. If you run the snake into itself, it's game over.

AstroLander

AstroLander is found in NeoTokyo on Normal or Hard mode.. It's in one of the lockers in the room with the SPB90 and the computer uplink.

In AstroLander, your main objective is to land a spaceship on a landing pad. However, you aren't allowed to crash, which is the main hazard in this game.

RetroRacer

RetroRacer is found in the Robot Factory in the final room before you fight the Machinist under a ramp in the northeast corner of the room.

The RetroRacer game is a lot like the classic Pole Position game in some regards. It's a simple racing game who's only objective is to complete five laps.