What are Mandalorians?
Above: No, he hasn't been born yet
The Mandalorians are a clan of diverse species that are bound by their culture. They have a long history of waging holy wars on neighboring planets, wearing the coolest armor ever, and being shot by blind smugglers. After Mandalore the Indomitable lost a battle to a Sith Lord named Ulic Qel-Droma, he forged an alliance with the Sith.
That's so Revan [KoTOR Spoilers]
(3963 BBY - 3956 BBY)
The Mandalorian Wars saw the rise of Knights of the Old Republic protagonist-gone-antagonist-gone-protagonist, Revan and his mediocre friend Malak. Some Jedis had a vision of genocide at the hands of the Mandalorians and granted Revan and his ragtag team of volunteers permission to become an official Jedi task force, called the Mercy Corps (which would be a good band name, by the way).
Above: That is, for the record, so Revan
Revan and Malak grew in power and their task force eventually joined The Republic, where they had immense triumphs on the battlefield. Revan spearheaded an amazing comeback against the Mandalorians, in which the nearly defeated Republic was able to once again subdue the Sith (who, as you may have noticed, lose pretty much all wars, ever). Unfortunately, Revan's moral fortitude suffered during this time, and he took merciless shortcuts that led to the destruction of certain "unimportant" planets.
Revan and Malak eventually traveled to the Sith world, Malachor V, where they were granted immense dark side powers, and fell from the light side. A year later, they would return as Darth Revan and Darth Malak (see what they did there?) to declare war on the Republic.
The Knights of the Old Republic
(3959 BBY - 3956 BBY)
The events of the Knights of The Old Republic spanned dozens of planets, battles, and years but the important factors were that 1) Revan was redeemed 2) Malak was killed, and 3) because of Revan, all but 100 Jedis were killed. That must have been a fun homecoming party.
The Sith Empire and Jedi Council were ruined. The Sith began to be conquered by neighboring colonies, and the Jedi Council publicly disbanded. All that was a few dozen people with lightsabers and Ugg boots.
Above: Remember the mask, it's really important (no it isn't)
Revan disappeared to the Unknown Regions (yes, in the Star Wars universe, Unknown Regions is a proper noun) to subdue a secret threat. He would never return.
Then began the Jedi purges, in which the remaining Jedis began to be weeded out by Sith, frequently a triumvirate of Sith Lords. Even more were killed, until a Jedi Exile (protagonist, KoTOR 2) was able to slay all three.
Time passed and the Jedis she trained were instrumental in reestablishing any semblance of a Jedi Council that will take place in Star Wars: The Old Republic.
The Great Galactic War
(3681 BBY - 3653 BBY)
The Great Galactic War serves as the backdrop to The Old Republic. It takes place 300 years after Revan disappeared. There are faux-peace treaties going on, an attack on Coruscant, and a small, but rebuilding, Jedi Council.
Is the Republic the same as the Rebel Alliance?
Above: Because the Senate is interesting
No, not really.
By the time of the movies, the Galactic Republic was long gone. The Rebel Alliance (which wears a few names, including The Alliance to Restore the Republic and Alliance to Restore Freedom to the Galaxy) are just a rebel group that supported its ideals. They stood in direct opposition to The Empire, which is closely correlated with the Sith.
Still not satisfied?
The background of one of the most complex fictional universes is a lot to cover in a single article. If you're looking for a more complete backstory, there's a webcomic produced by Dark Horse Comics - a quick read that fills in a few gaps in an excellent visual style.