Running and Backdash
In King of Fighters Running and Backdashes allow for alternate ways to move about on the screen at a greater speed. A run is executed by taping forward once, and then holding forward for the amount of time that you want the run to last.
Backdashes are executed by tapping backward twice.
Running
Running allows to greatly improve your ground speed, and runs make it possible to punish stuff that in other games would seem like the knock too far back to punish. It is therefore essential to quickly realize when something is punishable and immediately dash in to get in close and land your juiciest combo of a close hard attack.
After the run has started you hold down-forward as a direction and still continue to run. This allows charge characters to get a down charge while still moveing forward.
Running is also simply for covering ground space quickly without having to commit to a superjump or hyperhop. When you stop running there is a very short recovery animation in which you cannot do anything. Therefore you cannot go from run straight into block, which implies there is some risk to just running straight forward.
Runs can be cancelled at any point during the run into attacks, hyperhops, superjumps or rolls though. If you press a button for an attack you will stop in your tracks instantly and the attack will come out. If you tap any of the directions during a run, not a hop but a hyperhop will come out instead. Similarly, if you press and hold up in any of the directions during a run, a superjump will come out instead.
You can also choose to cancel the run into a roll, if you roll from a dash, you will roll slightly further than if you roll from a standing position.
Backdash
Backdashes carry absolutely no invincibility in The King of Fighters XIII, unlike many other games such as Street Fighter IV. During a short amount of startup, you will be airborne during the backdash, which does mean that you cannot eat a large combo as you will be reset. Often characters get a bit of invincibility at their feet during a backdash, which allows you to dodge some low attacks, but this is hardly a reliable means of doing so.
If certain characters have command normals that can be executed in the air, they can also be used in the airborne part of the backdash. For example Kyo can backdash and then instantly press d+C, and it will come out. The function of this, is that it shortens the hitbox of the character, which means it takes longer to hit the ground again, which means the backdash actually carries you further away. Other characters that can make great use of this are Iori and Athena.