Fate

Fate's character creation emphasizes it's unique narrative focus, and it involves the entire group in the process.   To start each player decides on a "high concept", such as "Wizard Private Eye" or  "Low-level Thug for the Syndicate".  While this is sort of like a class in another system, it's entirely up the player: there's no list of concepts.

The player similarly selects a "trouble" for their character, which is the central complication in their life, such as "Anger Management Issues" or "Don Giovanni Wants Me Dead". After that they just pick a name, and then start their first "adventure" ... by writing down a few sentences describing it, on an index card (or shared Google Doc).  They also gain another "Aspect" from the adventure , which is just like their concept/trouble (eg. if their life was saved by an NPC in their adventure, they might gain "Owes Life To That NPC").

Then the group rotates stories, and adds a complication involving their character to the adventure they inherited, again gaining an Aspect.  Then they rotate (in the same direction) and repeat the process, so at the end everyone has a first adventure involving two others, and they have five Aspects.

Next each character chooses a "pyramid" of Skills (ie. one skill at the highest bonus of +4, two at +3, three at +2, and four at +1).  While there is a fixed list of skills, groups are encouraged to modify it for their game.

Finally each character gets stunts, which are unique "tricks" the character has, similar to feats/advantages/edges in other systems.  Although the rules provide some guidelines, again the players must define their own stunts.  Characters can start with up to five, but players are encouraged to pick only one at creation and choose the rest later.

To accomplish things in Fate a player rolls a special set of four "Fate Dice", which have symbols indicating 1, 0, or -1.  Regular d6s could easily be used instead as long as you don't mind doing quick mental math (1-2 = -1, 3-4 = 0, 5-6 = 1).

The potential outcomes for any roll can therefore range from -4 to 4, and a character adds their skill bonus to this roll as well. If they can make a connection from the action to one of their Aspects they can either add +2, or get an opportunity to re-roll. The result has to beat their opponent's roll, or (if there is no opponent) a number chosen by the GM based on the action's difficulty.

If they win by two or more the action succeeds in style, giving the character an added bonus.  If they tie, the action still succeeds, but at a minor cost.  If they lose, the player can still opt to have the action succeed, but in the process they incur some kind of major cost.

As with many narrative systems, combat in Fate is just one type of "conflict" (other examples might include surviving an interrogation or a dangerous journey).  Conflicts mostly uses the game's same core mechanics, not a new set of rules (like Dungeons and Dragons).  

By default turn order in combat is determined by comparing the participants' Notice skill, and having the highest go first.  However, Fate has a variety of other optional initiative systems, ranging from "popcorn initiative" (where the GM picks the first participant, that player chooses who goes next, and so on) to simple GM fiat.

Combat itself is resolved as an opposed roll between the attacker and defender.  Damage comes in two forms: Stress (essentially "hit points") and Consequences.  Consequences represent serious injuries, and are essentially negative Aspects that the player gains.

However, in addition to simply attacking a foe, a character can instead use their action to provide a fellow character with a bonus.  For instance, one character might try to trip the opponent (providing a +2 to the next player who attacks them).

Because of Fate's rolling mechanic, a single difficult foe might have a much greater chance of being able to hurt the characters than they have of hurting it.  In such fights, instead of having every player try (and likely fail) to "chip away" at the foe's life (D&D-style), several characters might just focus on providing bonuses (eg. one trips the foe, while another throws something on their face to blind them) so that a final character has a greater chance of success.

Finally, it's important to understand that in Fate only combats that have a narrative reason should actually be played out.  An entire "fight" (in a game like Dungeons and Dragons) might be handled with just a single opposed roll in Fate ... if that fight is an unimportant one with more minor opponents.