What are the coins for?

When you run over a coin, you get a short burst of extra speed. Each coin that you get in a race, up to 10, will increase your Kart's maximum speed.

Your cumulative total of coins collected will unlock options for your Kart - including alternate Kart bodies, wheels, and gliders.

Coins will not grant a free life.

Unlike in the first Mario Kart, where they acted as a buffer against other players crashing into you, coins appear to act as currency now for buying new kart components.

Coins are to be found along a race track, and can be underwater, on the road, or in the air. Collecting coins on different tracks will allow the player to unlock vehicle parts used for customization.

Source: Mario Wiki