Time trials are types of combat racing events featured only in exhibition mode of Jak X. You are alone on the track, and the goal is to simply beat the highest score established either by yourself the first time through or by someone else if you are connected to the internet. As such, this event is used to challenge yourself and for practice, as well as to potentially establish a new universal high score.
Note that you can also toggle the "Use ghost racer" option in the "Advanced options" menu (though it is on by default), which will show your race car (or another's if connected to the internet) as it raced through the previous time. This helps in gauging where you will have to be to beat the previous time, as well as show you where you went wrong so you can correct it. The ghost car used will be what was used for the previous top-held time, regardless of whether or not you switch racers or vehicles. Ghost car eco pick-ups will not affect you as the pick-up will immediately respawn.
On that note, all eco pick-ups are absent aside from green eco for health and blue eco for turbo. As such, this event is all about speed. Pick your fastest race car, focus on powersliding, and try to turbo as much as possible (except at sharp corners). There are no medals, and instead you are simply trying to beat the high score.