Hopping is a technique that has existed in every Mario Kart game except Mario Kart: Double Dash!!, which removed hopping presumably because having two racers on a single cart made it too heavy. Hopping is performed by pressing and releasing the powerslide button, which has been the R button in every game to date. If the player hops while holding down a direction on the D-Pad or Analog Stick, they will start to powerslide.
While hopping will only give you a very small amount of height, it can be quite useful in the series. If timed properly, hops can clear small cracks or gaps in the track, such as those on Sherbet Land, which may allow the racer to shave off precious milliseconds from their lap times. When used in tandem with a Mushroom, hopping can clear much greater distances, allowing the player to use shortcuts that may save several seconds, such as the shortcuts found on Sky Garden. Hopping also allows you to avoid small bumps in the road; running over these bumps will cause your kart to bounce (which is bad, because it's usually slower to get air than to keep your kart on the ground).
Hopping can also be very useful in quickly changing the direction of your kart. If you tilt the Analog Stick in the direction you want to turn and hop repeatedly, your kart will rotate in that direction. This is very useful for situations where you are about to hit a wall and are going too fast to turn away quickly enough.