Can someone explain to me the story of Star Wars The Old Republic?

Answer: Okay with no spoilers... here is the basic premise of the game as your character starts:
Star Wars the Old Republic LEVELING GUIDE

This is about 300 years after the Knights of the Old Republic games. Where Darth Revan (a jedi who fought against the Mandalorians and then turned to the dark side and became a Dark Lord of the Sith) trained/led an army of soldiers and force users, trained in the sith traditions, against the Republic. The games dealt with the results of this war, and had foreshadowing of the return of the true sith.

The ancient Sith Empire has returned (They were thought to have been destroyed long long before this, but a fragment of the Empire had been hiding on a remote planet and rebuilding.) and attacked the Republic in a massive battle and fought them to the point of a forced peace treaty between the two groups.

The Jedi are not in exile, they are still part of the Republic though they have had to move their training of new jedi knights and consulars to an older world that used to house one of the major jedi temples. The Sith have retaken Korriban and are using it as a training world for new sith warriors and inquisitors.

The game centers around your characters making decisions on how the results of this tentative peace effect you. Whether you act to protect the treaty or actively work to break it and return the galaxy to war. (And yes there are groups on boths sides, Republic and Empire, that are trying to restart the war.) And each class has a separate class story arc that starts at level one and goes all the way to the end game.

Star Wars the Old Republic LEVELING GUIDE