System 101
This page gives a brisk rundown of the mechanics of KOFXIII.
Rounds
The King of Fighters is a 3 on 3 fighting game like Capcom vs SNK 2 where each teams' characters fight each other one on one in order of each players' selection. A full match can have from 3-5 rounds.
The winning player receives a health restoration bonus for the next round.
Timer
Each round in XIII lasts 60 seconds.
Life Meter
The green bar located at the top of the screen. All characters have identical life amounts.
Guard Meter
Found underneath the health bar, this blue meter depletes each time a player blocks a normal or special attack. Emptying the bar results in a guard crush where the player is unable to act and can be freely hit for a few seconds. The bar begins to regenerate after a few seconds when not blocking.
Stun
Stun is a hidden value that cannot be viewed. Stun builds when being hit and a player will begin to lower one's stun amount once on the ground once back to neutral.
Once stunned, a player can attempt to mash the buttons and directions to recover faster. A stunned player is subjected to damage scaling from the previous combo that lead to the stun.
Super Meter
Super Meter is represented by the orange bar on the bottom of the screen. It's used for Desperation Moves (Supers), EX Specials, Neomaxes (level 3 supers), Guard Cancel Blowbacks (alpha counters), or Guard Cancel Rolls. A player can store up to three meter stocks on their first character plus one additional level for each teammate K.O.'d for a total of five meter on the anchor (last) character.
Meter can be built by using specials, hitting the opponent, being hit by the opponent, or by blocking attacks from the opponent. Meter will also carry over after every round.
Hyperdrive Meter
The smaller cyan bar above the Super Gauge. The player can use 50% of the bar for a Drive Cancel or Super Cancel, or save the full bar for entering Hyperdrive Mode or to use a Neomax.
The Hyperdrive Gauge builds in conjunction with the Super Gauge from the same set of criteria.
Drive Cancels
Drive Cancels allow a player to cancel specific Special Moves directly into other Special or Desperation Moves.
Simply input the command for another Special or Super during one Special and if the player has half of their Hyperdrive Gauge filled then the cancel will occur. Not all Specials can be canceled and other are a poor choice to cancel into; play around to see what works or simply check out each character page for optimized usage.
Hyperdrive Mode
The most powerful use of the Hyperdrive Gauge; leads into the heaviest combos in the game. Hyperdrive Mode allows the player to cancel a normal or command normal by pressing B+C and causes a character to automatically run forward with enough frame advantage to combo into a heavy attack.
Every normal and command attack becomes cancelable in this mode. In addition to this, the player can use multiple Drive Cancels in the following combo. Optimized Hyperdrive combos deal huge damage and can be concluded with MAX Cancels--the act of canceling a Super into a Neomax.
This mode lasts until the length of the Hyperdrive Gauge naturally empties over time, but each Hyperdrive Cancel will take a portion of the bar away. Using a Neomax will immediately terminate this mode.
Guard Cancel Blowback Attack
An attack that can be done only when in blockstun. It costs one bar of Super Meter to use and pushes the attacking player backwards into a soft knockdown but deals miniscule damage. These are mainly used to get out of a bad situation or to get another chance to work the neutral game, but the meter cost adds up.
It's also possible to bait and counter these attacks.
Rolling
By holding forward or backward and then pressing A+B, the player will roll on the ground. The startup of a roll is invulnerable to attacks up until the recovery at the end, but the rolling player is always vulnerable to throws.
For KOFXIII the amount of time a roll stays active has been increased which in turn increases the risk of rolling even more so than in previous games.
A successful roll can potentially lead to a full combo, although the heavy risk of being punished should limit the frequency of relying on them to land damage. Think of a 'random' roll as a jump-in from Street Fighter; if it works well it can turn into a big combo, but good players will confirm most rolls into guaranteed punishes. A player's main strategy should not involve landing a risky roll to win, just as a Street Fighter player shouldn't bank on landing a jump-in combo.
Guard Cancel Rolling
Similar to a Guard Cancel Blowback Attack; the player cancels blockstun into a roll that is completely attack and throw invulnerable. This costs one Super Meter and is mainly used to escape predictable blockstrings and possibly punish the opponent with a combo.
While the roll itself is unpunishable, the rolling player can waste a meter only to be put into a meaty/throw/crossup mixup if the opponent used a safe blockstring.
Hops, Hyper Hops, Jumps, and Superjumps
Covered greatly here.
Running
Activated by pressing 
, and can be continued indefinately by holding /
after starting the run.
Runs can only be canceled into attacks, forward hops/jumps/rolls or by stopping, which has a delay. Running is great on the offensive and can also be used defensively to anti-air the opponent by running underneath an aerial opponent.
Backdashing
Backdashes have no invulnerability, but are able to quickly move the player backward out of the reach of an offensive opponent to avoid a disadvantageous situation. They also can be canceled into aerial command normals and specials.
Throwthing
Throws in KOFXIII are back to being activated with one button and thus lack a whiff animation. When close to the opponent, hold /File:Back.gif and press 
/
to attempt a throw. If the opponent is unthrowable, a Close Heavy normal will come out instead.
Throw techs are executed likewise, but a successful tech requires pressing the same button the opponent initiated the throw with. This can be done by pianoing ~.