From The TimeSplitters Wiki

A Medal or Trophy is a reward for your performance in an Arcade League or Challenge scenario.

Types

There are four types of Medals, depending on how well you did on the particular challenge. Different medals give you different rewards. The Medals are as follows (in order from worst to best):

• Bronze - You get this if you pass the challenge, no matter what. These usually give 0-1 rewards.
• Silver - You did better than a bronze. With these you can unlock around 2 things.
• Gold - Very good. If you get one of these, you get all the unlockables for that challenge.
• Platinum - Excellent! If you get a Platinum, you get all the unlocks of the challenge and bragging rights. If you get these on EVERY challenge in the game, you are a true master (although be careful, as in TS2, as there is an unfixable glitch!)

In addition to having different names, each type of medal has a different image shown once you win it. A Platinum, for example, is a large sculpture of a TimeSplitter with a gun standing on a dead TimeSplitter, while a Bronze is a smaller, bronze TimeSplitter.
You can (almost, see Glitch) always improve your medal. If you get a Bronze on a challenge, then redo it and get the requirements for a Gold, you'll get a gold.

Unlocks

Depending on your medal, you get different unlocks. For example, in Can't Handle This, you get Chasm for a bronze, Hatchet Sal and Chasm for a Silver, and a Gold will earn you those two and the Handyman. Remember, you can always get the other unlocks by getting a better medal.

Glitch

In TS2, there is a glitch in the Banana Chomp > Playing with Fire challenge. If you get a Gold Medal, and save the game you will not be able to get a Platinum on that profile, ever. The only way to do so is to turn off your system before the game can save your Gold medal. Restarting won't help, you have to turn off the system.