One type of Running is just what you do to get around Tibia, it's sometimes also called walking.
Another type of Running in tibia is a style of fighting some vocations may use so they do not take damage. I will also explain this below.
Mouse (game window)
Most people use this way when they first start Tibia, you click in the game wherever you want to go and you go there. This is good if you want to get to an exact square. When using the mouse to move, you will reach your destination slightly faster than using the keyboard because of the relay time while the server waits to receive another command from your keyboard.
To use this travel method, simply click on your mini map (Top right of game) and your character will walk to wherever you clicked. This is a great way to travel between cities because it requires little interaction while you are walking.
When using the minimap, Tibia will try to find the quickest route (not necessarily shortest). If you do not have much of the map revealed, or if there are obstacles in your way, this may cause you to walk in the wrong direction, or to take very long routes. Reveal most of your map to make this method of walking more efficient.
One simple way of running is using your arrow keys to get around, The up key goes North (^), down key goes South (v), right key goes East (>), and left key goes West (<). This is the easiest way to walk in some peoples opinions, and is usually used when fighting monsters to avoid their wave/beam attacks, or to position yourself to aim and attack them with spells.
Num Pad Running
This way isn't used as often, but you can walk using the Number Pad (right of keyboard). This is the only way you can walk diagonally without items being in your way. If you have NumLock on, you have to hold the Shift key (usually displayed as an arrow pointing up), then when you press the NumKeys: the 1 runs Southwest, 2 runs South, 3 runs Southeast, 4 runs West, 5 cancels all actions, 6 runs East, 7 runs Northwest, 8 runs North, and 9 runs Northeast. If NumLock is off, you can do this method of running without holding the Shift key.
This type of running can be used by any vocation besides Knights (can be used by knights, but is not most of the time because they are a strongly melee based vocation.) This type of running can keep most vocations from taking very much damage.
What you do is if you have any distance weapon or rune, you can run away from a monster with it following you, keeping far enough away so that it cannot attack you, and close enough so that you can still attack it. Some monsters are very fast, so they are harder to run unless you have a higher level.
You can run straight in a line away from the monster, or if there is a cluster of bushes or rocks, you can run around the cluster and shoot at the monster. Same as before, be within distance of shooting and far enough so you aren't hit. If you run around a cluster, sometimes the monster will turn around as it is a shorter distance for the monster to walk the other way to hit you. This isn't a major problem, but it can be if the monster is very strong, so be aware not to go too far away if running around a cluster.
Here is an example of what I was showing above.
X M X XXXXXY X X
This is an example of a famous way to run Giant Spiders in one area of the Plains of Havoc. X is the cluster, M is the monster and Y is you. (you are walking south (down), and the monster is walking north (up)). Nothing else really to explain here, this is just a diagram of what running around a cluster may look like.
Be aware if you are running, some people may get in your way in attempt to kill you if you run monsters, so keep that in mind.
Also, not all monsters can be run. Mainly ones that have distance attacks. 2 examples of good monsters to run are Cyclops and Giant Spiders (GS are fast, so higher levels are the ones who can run these.)