Running and Backdash
In The King of Fighters, Running and Backdashing allow for alternate ways to move about on the screen at a greater speed. A run is executed by tapping forward twice, 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, or dashing, is obviously mainly used for covering ground space quickly. Because your ground speed increases so much, it has the advantage of being more difficult to react to, and it can let you get in without having to commit to a superjump or hyperhop. However, when you stop running, there is a very short recovery animation, during which you cannot do anything. Therefore, you cannot go from run straight into block, which means 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. If you tap an up direction during a run, a hyperhop will come out instead of a normal hop. 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.
It is important to note that when pressing a heavy attack out of a run, you cannot get a throw. This means that dashing up and hitting C or D will always result in an attack coming out. This particular trait is very helpful to some characters, such as Kyo and Iori, whose close C's are very fast, have good activation range, and angle upwards to catch jumps. This makes running forward and pressing C appropriately a powerful pressure tool for characters like this.
After a run has started, you can actually hold down-forward and still continue to run. This allows charge characters to get a down charge while still moving forward and can let you instantly go into important moves such as a crouch B for a low or a crouch C to anti-air.
Remember, if a move seems unsafe on block, but you can't seem to get close enough to punish it, try using runs! The extra distance you can cover during it will allow you to punish many moves that would appear safe in other games, netting you a big combo from a juicy close heavy attack. It is very important to know about moves you can punish like this, so take note when you can.
Backdash
Backdashes carry absolutely no invincibility in The King of Fighters XIII, unlike many other games such as Street Fighter IV. Immediately upon execution, you will be airborne during the backdash, which does mean that you usually cannot eat a large combo as you will be reset. Characters often get a bit of invincibility at their feet during a backdash, which makes it possible to dodge some low attacks, but this is hardly a reliable method to avoid lows. In The King of Fighters, as a general rule, Rolling is more often used for the same purposes as invincible backdashes in other games, as it has invincibility and lets you escape from more dangerous situations, and backdashes are more about creating space in a neutral situation.
If certain characters have command normals or specials/supers 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 the move will come out as he is airborne during the backdash. This shortens his hitbox, which means it takes longer to hit the ground again, resulting in the backdash actually carrying him further. Other characters that can make great use of this are Iori and Athena.
Aerial specials during backdash are more tricky to define. There are various uses of them but they all really depend on the character. For example, Benimaru's Raijinken (Air qcf+P) will float him slightly and has a lingering hitbox, allowing you to use it to cover your backdash with a projectile.