Rifts are a kind of activity in Saints Row IV, with the objective differing depending on which kind of Rift you are doing. There's three different types of Rifts in the game, which are Speed Rifts, Telekinesis Rifts and Platforming Rifts. Telekinesis Rifts have you throwing red/yellow spheres at the correct colored spots, which can change to the other color every so often.
Speed Rifts will have you sprinting (you automatically do this) while collecting spheres for points, as well as dodging obstacles. These spheres will fill up a meter that can be used to activate a power that allows you to hit the obstacles and get points from those, too.
If you remember the Platforming Rifts story mission, then you need to jump from platform to platform. Where you land on the platform determines how many points you get, with more being awarded the closer you get to the middle. There's some platforms where you need to sprint on them to get the points. The activity will end when you either run out of time or land on the final, big platform.