A time trial is a concept where players race against the clock to get the fastest time possible. It allows the player to drive around the course all by yourself to get the best time on the track. In the Mario Kart series, players use 1 to 3 mushrooms to help the player during your run and this mode is a great place to unlock new vehicles and characters like Birdo and Toadette.
Time trials can improve or hinder the enjoyment of any particular game and it comes down to what the player enjoys, some hate it and some love it. Some games have an unlockable time trial mode upon completion of the main story. Attempting to beat times set by players on leaderboards lengthens the life of a game often to the point of surpassing the original story mode. One such example of this is Mirror's Edge, the campaign was around 5-6 hours in length but to pass all the time trials requires the player to dedicate much more time.
A semi-recent phenomenon is the time-trials only game. Games in the category include the Trackmania and Trials game. Games often use this when racing against actual people or bots would hinder on the games enjoyment. For instance, it's hard enough getting through a Trials HD track on it's own so they'res no real point in having any more competition.