In the end of Return of the Jedi, all of the known Sith are dead and utterly defeated. So how did Snoke, who looks very old, manage to turn Kylo Ren to the dark side? And Snoke couldn't be alive at the time of Darth Sidious because of the Rule of Two. I mean, is it possible for the dark side to just appear in someone?
The dark side of the Force is a continuously existing aspect of the Force. It exists whether or not there are any Force-sensitive individuals who use their anger for their power (i.e. the "dark side").
There are dark side practitioners who are not Sith, such as the Inquisitors, Nightsisters, and the Knights of Ren. Thus, the apparent extinction of the Sith does not mean that the dark side of the Force no longer exists nor that there are no living masters of the dark side.
We have no confirmation that Snoke is a Sith, so he apparently belongs to some other dark side organization -- either one we already know about or a new one. We don't know how he became skilled with the dark side, but he learned how to use it somehow and that's how he was able to seduce Kylo Ren to the dark side.
Any Force-sensitive individual can become skilled with the dark side of the Force. The first Sith were Jedi who rebelled and turned to the dark side.
I believe this Rule of Two is often taken too literally.
And Snoke couldn't be alive at the time of Darth Sidious because of the Rule of Two
Although the Sith follow some sort of philosophy, you can't imagine that a selfish, murderer, insidious, cold hearted Dark Lord would feel obliged to obey some rule.
Take for example the theory that Snoke could be Palpatine's former master, Darth Plagueis. If he was indeed hiding in order to make Palpatine think he was dead, can you imagine him deciding: "Oh, well, Palpatine just took an apprentice... Damm, now I've got to stop hiding because I cannot disobey the Rule of Two.
Of course we can make no assumptions before Episodes VIII and IX. Neither can we be sure that he is indeed a Sith. But even if he is, that doesn't mean he was not around during or before the events of Return of the Jedi, or even longer than that.
The Sith and the Dark Side
Two separate things. All Sith are dark side, but not all dark siders are Sith. As mentioned in the other answers, and answers to other questions, Kylo Ren is neither Sith nor Jedi. Snoke may or may not be a Sith; we don't know yet. But we have seen hints that Snoke isn't a Sith either:
The Supreme Leader [Snoke] believes Ren to be the ideal embodiment of the Force, a focal point of both light and dark side ability.
- The Visual Dictionary
The previous leader of the Sith, Emperor Palpatine, seemed to be devoted exclusively to the dark side, so by comparison, Snoke doesn't seem to be very Sith-like at all.
The Rule of Two
Not so much a rule as it is an observation of the way things usually work. We know that Darth Tyrranus (Count Dooku), Darth Maul, and Darth Sidious were active at the same time, so there is canonical precedent for the "rule" of two being broken.
The Idea That The Dark Side Ever Went Away:
In the Star Wars movies, no one ever says that the dark side is the Sith. Quite the contrary, in fact. The dark side is variously associated with:
Lust for power
None of these things ceased to exist when the Empire was destroyed, and chances are, none of them will ever cease to exist. The dark side didn't come back, because it never went away. These things are insuperable parts of human experience. The Force Awakens isn't about the return of the more regrettable aspects of human nature, it is about the rise to power of people consumed by these negative aspects of human nature.