Made this community wiki for anyone to fix up. I'm trying to keep within the limits of the new canon as much as possible:
The Galactic Republic
The Galactic Republic was the largest single collection of planetary systems prior to the formation of the Galactic Empire. Each system was represented by Senators, and certain other collectives/organisation had some representation (for example, the Trade Federation). It was bordered by a number of independent systems and other collectives (such as Hutt Space on the Outer Rim).
About 30 years before the Battle of Yavin (the date of the New Republic calendar), the Republic become embroiled in a decade long civil war. While it appeared to win this civil war, the Republic was dissolved into a Galactic Empire under Emperor Palpatine (who is also a Sith Lord).
(In The Old Republic game, the Galactic Republic formed about 1000 BBY out of what became known as The Old Republic).
The Jedi Order
The Jedi order formed thousands of years BBY - it consists of Force users primarily focusing on serving the common good.
At various times, the Jedi would enter into open conflict against the Sith Order. The Jedi also militarised to support various governments - particularly in the Galactic Civil War in 30-20 BBY in support of the Galactic Republic.
The Jedi Order was wiped out around 20 BBY by the new Galactic Empire, with almost all survivors being hunted down and killed by 0 BBY.
The Sith Order
(Note - in the EU, the Sith were a race of force-powerful beings - eventually, they died out to be replaced by an Order of Sith).
The Sith Order was diametrically opposed to the Jedi - seeking personal power, and giving in to their passions, opening them to the Dark Side of the force.
Eventually, the infighting within the Sith Order caused the Sith to almost completely self-exterminate. This brought about a Rule of Two, which continued up to the death of Palpatine and Vader in 5 ABY.
The Sith were responsible of the transition of the Galactice Republic into the Galactic Empire.
The Trade Federation was an interstellar shipping and trade conglomerate in the Galactic Republic. It had considerable control over the trade routes and markets within the Republic.
The Trade Federation were manipulated by the Sith to become one of the founders of the Confederacy of Independent Systems - a separatist movement which entered into the civil war with the Republic which ultimately led to the formation of the Empire.
Confederacy of Independent Systems (Separatists)
The Sith manipulated a number of worlds and organisations into declaring independence from the Galactic Republic sometime between 30-20 years BBY. The major founders were the Trade Federation, the Commerce Guild, Techno Union, and Intergalactic Banking Clan.
Eventually, as the Galactic Republic reformed into the Galactic Empire, the Sith ended the civil war by assassinating the leaders of the Separatist factions.
The Galactic Empire formed from the Galactic Republic in 20 BBY. It isn't clear when it officially dissolved completely, but the Emperor was killed in 5 ABY and a large portion of the Empire became the New Republic at this time.
It also seems the Empire might have extended its influence into more systems (as we see them garrison worlds like Tatooine).
The Rebel Alliance
Sometime between 20 BBY and 0 BBY, there were a number of worlds that were dissatisfied with the dissolution of the Republic into Empire. Eventually, these worlds formed an Alliance which grow into a vast military capable of combating the Empire in daring raids and strikes. Eventually, in 5 ABY, the Alliance succeeded in killing the Emperor and defeating a significant portion of the Imperial Fleet.
After a number of years, the Alliance successfully restored representative government and formed the New Republic. However, this New Republic was smaller as a number of worlds were still held by Imperial forces.
The First Order
Holding onto a portion of what was once the Galactic Empire, as of around 35 ABY there seems to be an uneasy peace between the New Republic and the First Order - particularly as there is a Resistance within First Order territory which is supported by the New Republic.
The Knights of Ren
Very little is known of this group - so far, two members have been revealed: Kylo Ren and Supreme Leader Snoke. They appear to be force-users and the group may have developed from Sith teachings (as they appear to tap into the Dark Side of the force), but their origins and motivations are unclear at this time.