As you mentioned, the snow owls are linked with the lemmings. This is true of any predator-prey population model. If there aren't many lemmings (prey), then the owl population will go down. This is due to lack of food. The unlucky members will starve and die off. As the predator owl population decreases, the lemming population will go back up. This is because the lemmings aren't being eaten by the predators (as there are less owls to be eating the lemmings). Then as the lemming population goes up, so does the owl population.
As one population goes up, the other does too. This is until they reach the carrying capacity and the populations go down. This cycle repeats itself. It's quite common to find populations in flux all the time like this. Keep in mind that we're only focused on the food aspect and not other factors such as habitat. If the lemming habitat is severely compromised, then that will also affect the lemming population. To keep things relatively simple, we'll assume the habitat stays constant.
Side note: The common misconception is that lemmings jump off cliffs on purpose, and kill themselves, but this is a myth that was unfortunately spread around. Lemmings do not do this. If they did, then the species would have died off long ago.