FATE Accelerated Edition, or FAE, is a good introduction to Evil Hat's FATE game engine. While it's different enough from FATE Core to be a stand-alone RPG in its own right, FATE Accelerated is similar enough to its parent so that once you feel you're ready to tackle the more advanced mechanics of FATE Core, you should be able to make the transition with little effort.
FAE is extremely liberating from other RPGs, in that there are no Character Points to spend, no endless lists of skills, powers, talents, traits, or extras to choose from, no defects/disadvantages/flaws to burden your hero or heroine with just for the sake of balancing out a character's point totals. Character creation is simple and easy: you choose a "high concept" (a phrase that sums up your character's role in the game setting), a "trouble" (another phrase that identifies a primary weakness of the character), and one to three other "aspects" (additional concepts that help round out the character's background). You then set six "approach ratings," which represent how your character deals with various challenges (Cleverly, Sneakily, Forcefully, and so on). If desired, you can choose anywhere from one to three "stunts" (things your character can do extremely well). In a matter of minutes, you have a fully defined character with a general suite of abilities which have been established by his or her approach ratings, high concept, trouble, and other aspects. All without the numbers crunch you have to deal with from other RPG systems.
The task resolution system follows the general path of "roll-high-and-hit-a-Target-Number," only in FAE, you roll four Fudge/Fate dice and apply the result to whatever approach rating you're using for a particular task. Having an appropriate stunt can improve your chance of success, as can the expenditure of Fate Points. If you're trying to accomplish something that fits your character's high concept (or other aspects), you can spend a Fate Point, which either gives you a bonus to your roll or allows you to roll again (this second option is best if you initial roll was very poor). You gain Fate Points during the game by accepting "compels," which is when your character's adversaries find a way to use your trouble (or even your other aspects) against you; if you give in to your weaknesses, you're rewarded with a Fate Point, which is a great way to encourage good roleplaying.
Combat is quick and easy to run. Instead of worrying about how much damage this weapon does against that sort of armor, everything is resolved with a single pair of attack/defense rolls; the greater the attacker's margin of success (or "shifts"), the more damage done to the target. A target can stave off defeat by checking off stress boxes or accepting "consequences" (penalties which give the attacker additional weaknesses to exploit); when the target can take no more stress or consequences, he or she is at the mercy of their attacker. A neat option allows a would-be victim to escape this fate by voluntarily conceding the fight before the coup-de-grace is actually delivered, allowing them to exit the scene with at least a modicum of dignity--and they also get a Fate Point for doing so.
Overall, the system is very free-form, with all kinds of room for innovation by both the players and the gamemaster. This is arguably the most cooperative RPG system ever created: everyone works together to weave an exciting narrative, using aspects, troubles, and the like to take an adventure in all kinds of unexpected and enjoyable directions. This can be a bit daunting for gamers who are used to more rigid systems, but once you get into the spirit of the thing, you'll find yourself having more fun than you could have imagined.
FATE Accelerated Edition is a terrific game, and definitely worth checking out. The list price for the published edition is $5.00; I recommend paying something in that neighborhood for the electronic edition, as it's well worth it!