Shadowrun

To create a "shadowrunner" a player spends 400 "BP" (build points). As a point-based system there are no traditional classes: the game provides templates like "Bounty Hunter", "Hacker", and "Radical Eco-Shaman", but they are just pre-designed builds, and players are free to create any kind of character they want. 

Metatype and Qualities

To begin they select a "metatype" (ie. race) for the character.  The core book includes options for humans, elves, dwarves, orks, and trolls, and every race except human provides special benefits, while also costing extra BP (eg. Trolls, the most expensive, cost 40, while Dwarves are only 20).

Next, the player can buy "Qualities", which are a bit like Advantages/Edges/Feats in other systems, as they provide benefits such as having animal empathy, being lucky, or being a magician (with each Quality having its own point cost).

Attributes

After that the player buys the character's attributes.  There are eight attributes; essentially the D&D set, except Dexterity has been split into Agility and Reaction , and Wisdom has been split into Intuition and Willpower.  Each attribute point costs 10 BP, except for the final point in a stat which costs 25, and only 200 points can be devoted to attributes.

There's also two special attribute, Edge (ie. luck, which can be improved) and Essence (the character's life force; it can't be raised, but like Humanity in Cyberpunk, it can be lowered if the character gains cybernetics).  Magical characters also have a "Magic Resonance" attribute, and every character has a (derived) Initiative attribute.

Skills

Next, the player buys skills, which rank from 1 to 6, with each rank costing 4 BP.  There a fairly large number of skills to choose from, each of which has an associated attribute.

There are also "skill groups", which cost 10BP instead (eg. instead of taking Longarms, Pistols, and Automatics, a player could buy the "Firearms" group), and "specializations", which cost 2BP, or 1BP for knowledge skills (eg. a character who wants to be better at shooting revolvers in particular could buy a specialization to gain two extra dice when shooting them ... but not with any other pistols).

Resources

Finally, the character needs to assign their resources.  This will include purchasing gear (weapons, armor, and non-combat), cybernetics, or magical spells,  adept powers, technomancer resources, or contacts (ie. acquaintances).  Again, all of these can be purchased with BP (eg. 1 BP provides 5000¥ of gear, while spells cost 3 BP each).

Once that's complete, they simply have to decide on their background story, and they're ready to play.

Most rolls in Shadowrun are skill checks, which involve rolling a number of d6 equal to the character's rank in the particular skill, plus their rank in the corresponding attribute.  For instance, a character attempting to sneak with an Infiltrate skill of 4 and an Agility attribute of 3 would roll 7 dice.

Hits

Every die which rolls a 5 or a 6 is a "hit".  Many actions only require a single hit to succeed, but others have a "threshold" (which the GM is encouraged to keep secret) which requires as many as 4 or even more hits to succeed.  Many skills have specific tables detailing how many hits are required to use that skill (using escape artist to get out of ropes is a 4, but escaping from a straitjacket is a 6).  If a character scores extra (unneeded) hits, the action succeeds with extra flair.

If the action isn't immediately risky a character can exchange rolling 4 dice for a single automatic hit, but a character has to either buy or roll all hits; they can't mix the two.

Glitches

If half of the dice rolled or more are 1's, the action "glitches": even if it succeeds, some other negative consequence results (eg. the character might succeed at jumping over a table, but knock over a beaker of acid on top of it in the process).  If a character glitches and scores zero hits for the roll, it becomes a "critical glitch", and something really bad occurs instead.

Occasionally no skill will apply to a roll, in which case the GM will choose an attribute (or often two) to use for the roll instead.   For instance, maintaining one's composure when faced with a monster is a Will + Charisma roll.

Edge

Finally, a character can spend an Edge (from a pool that's based on their Edge attribute) before rolling, to add a number of dice equal to their Edge attribute to the roll.  If those dice roll 6's, those 6's count as two hits instead.  Alternatively a character can spend Edge after a failed roll, but in that case 6's only count as a single hit.

In Shadowrun combat order is re-determined every round.  This is done by having every participant roll their Initiative attribute, and then they add that attribute to the number of hits rolled. Edge can be used on the roll, or one Edge can be spent to guarantee first place ... unless another participant also spends an Edge.

Many characters in Shadowrun can take multiple actions in a single round of combat.  Such characters get to take an action as normal on their turn, but then after everyone else has gone they get a second (or possibly 3rd-plus) "Initiative Pass".  Since being able to act multiple times before others can is extremely powerful, having multiple Initiative Passes is extremely desirable.

Actions

On their turn a character can take a single Free action and either two Simple actions or one Complex action.  An example free action would be dropping something, a Simple action would be firing a weapon, and a Complex action would be firing an automatic weapon or reloading one.

Characters can also walk (their movement rate) for free, or they can spend their Free action to run instead.  Either way, if a character moves they're considered to be moving for the entire rest of the round, up until they next act.

Attacks

Attacks are made using the appropriate weapon skill (and corresponding attribute), but instead of needing a fixed threshold the attacker needs to beat the defender's roll, which is a pure Reaction check unless the character took a Full Defense combat option (in which case they can add their Dodge or Gymnastics skill), although they might get bonus dice from things like cover.  There are also various special rules for different types of weapons. eg. automatic, shotguns, etc.

Damage

If a character is hit they take either physical or stun damage, depending on the weapon used.  The defender makes (another) roll, of their Body attribute plus their armor's DV (Damage Value), and each hit reduces the damage taken by one.

If damage gets through it is applied to the character's appropriate damage track (either physical or stun).  Each player has 8 + 1/2 their Body physical, and 8 + 1/2 their Will stun: once either track is filled the character goes unconscious.  However, after the first three points (and each further three points) the character takes a cumulative -1 wound modifier to most rolls the character makes.