As of this WU there are two "Star Wars RPG" systems; the original Star Wars D6 (SWD6) by WEG and the more recent Star Wars D20 (SWD20) by Wizards of the Coast. Although both systems utilize the same core material, their game mechanics are almost completely different.
Star Wars D6
In October of 1987 West End Games aquired the license for the Star Wars franchise, and created a wildly successful RPG out of it. The system was revised into a 2nd edition in October of 1992, and finally into 2nd Edition: Revised and Expanded in August, 1996. After publishing nearly sixty books, West End Games went bankrupt and sold the franchise. As a result all SWD6 materials are out of print, but can still be found on Ebay. Do a search for "Star Wars D6" or "WEG Star Wars" to find what's available. There are also a couple PDFs available for some of the more common books if you scour the file-sharing networks.
Star Wars D20
When WEG gave up their license for the Star Wars franchise, it didn't take long for Wizards of the Coast to grab it. In November 2002 they released the Star Wars Roleplaying Game (henceforth called Star Wars D20), which is based off of the D20 system initially used in Dungeons and Dragons 3rd Edition. This was quickly followed by several sourcebooks, and (in the typical tradition of WOTC) a revised rulebook less than two years later in May 2002. Some 20 books and products have been released thus far, although several of the earlier sourcebooks (most notably Starships of the Galaxy) are awaiting updates to the more recent version. Like most WOTC products, Star Wars D20 is available at most major book retailers.
Star Wars D6
SWD6 is a skill-based RPG. This means there are no classes or levels, in a manner similar to White Wolf's D10 system. Instead players assign points into specific skills that best fit their character. A Jedi might have a hefty amount of skill points devoted into lightsaber, while Grandma might have devoted the same number of points into crochet1. These points show up as a number of dice plus some form of modifier (for example, 5D or 3D+2), and represents how skilled you are in a particular task. Progression is very slow and smooth, as the character improves one skill at a time.
When you want to perform an action, you either make an opposed roll or roll against a difficulty level. Opposed rolls are for when you perform an action against someone, for example shooting a stormtrooper. Rolls against a set difficulty are for unopposed events, such as picking a lock or flying through the superstructure of the Death Star. Difficulty ranges from 5 (very easy) all the up to and beyond 30 (heroic).
One of the nicer things about SWD6 is that you only need D6s. You need a lot of them, but you only need one kind.
Star Wars D20
Taking the opposite approach to SWD6, SWD20 falls back on the class and level based system made popular by D&D. Checks are based off the roll of a D20 that is modified by character attributes and abilities, as well as an circumstancial modifiers such as the quality of equipment being used. Higher level character have higher modifiers for abilities and attributes, and progression is typically very linear and regimented. Different classes are better at different things, and players can multiclass to choose a style that roughly fits what they want. At higher levels, more advanced classes called prestige classes provide even more flexibility.
Rolls are still made against either a set difficulty or by an opposed roll, but there is significantly less randomness in D20 relative to D6. Certain player abilities eliminate dice rolling entirely (for example, the scout class has the ability to make any roll for certain mastered skills automatically be a ten), turning the system in more of a numbers game. Combat is done through a series of rolls against set difficulties called the defense; this is done for both normal combat as well as vehicular combat. Oddly enough this means that piloting skill plays a very minor roll on whether or not you get hit.
Despite the name, SWD20 actually requires every common kind of dice out there: D4, D6, D8, D10, D12, D20, and D100.
In the end, comparing SWD6 to SWD20 is like comparing apples to apple juice. It's basically the same thing, just presented in a different way. I personally prefer SWD6's skill-based system for several reasons, most notably the flexibility allowed by not having to fit the mold of a specific class. Although it's more complex than D20, the additional rules are generally located in places where they need to be (such as starship combat). In addition, the extraordinary amount of material (both official and fan-made) provides a much greater foundation to work from with the D6 system.
It should also be noted that SWD6 is purely "classic" Star Wars, and is therefore untainted by certain modern unpleasantries.
1 Okay, there isn't really a crochet skill in SWD6. For that you'd have to go to GURPS.