Dodge

When you're diving away from gunfire or ducking under a baseball bat, it's all about reflexes and senses.

The ability to get the hell out of the way when someone's trying to hurt you.

This skill is used automatically every time someone takes a swing or a shot at you. The higher it is, the more likely you are to avoid the attack. This is doubly important when being attacked with ranged weapons, since it's your only defense other than armor.

But I never see myself dodging!

Active dodging only happens if you can't parry the incoming attack. With melee-vs-melee, much of the time you will never see a dodge happen. However, this doesn't mean your dodge skill is useless! Dodge is factored into every hit right at the start, so a high dodge skill makes you harder to hit in general, even if you don't see a message about it.

Mechanics
For every two dodge totals after 10 total you have will also contribute to a general (-1 tohit) debuff which is automatically used against your opponent's attack rolls. This can make it even easier to dodge and parry your opponent, or even make them completely miss their attacks if it is raised high enough. Because many weapons scale their damage on tohit; having high dodge can indirectly decrease the power of melee attacks by a large amount. This debuff is halved against guns.

Dodge allows you to negate 100% of damage, as long as it works. It is also the skill that allows you to escape from combat. Because of this, it is very hard to grind. Most players opt to just buy raws in dodge, instead of grinding it to 6 raw, like other skills. Dodge is also one of the most expensive skills to buy raws in, buying your 15th raw in Dodge may cost more than 12,000 EXP!

You can actively stop attacking and focus on dodging in combat for a couple of seconds by typing in combat which will raise your total by 5 dodge. It is not required to use this command to dodge, as dodging mid-combat is done automatically. It is useful when tanking for a group against a powerful enemy, an incoming power attack, or just buying yourself some time.

Addendum
As of April 2016 your weapon's total weight (rounded up) will reduce your dodge total in proportion. A 4.2kg weapon will reduce your dodge by a total of -5. This also applies to the total weight of dual wielded weapons.