Evasion is a defensive attribute, and affects a character’s ability to dodge incoming physical attacks. Evading a physical attack is the only way in which to avoid the damage all together. Characters with higher Evasion attributes are less likely to be hit by physical attacks, and thus will take less damage from physical attacks in general.
Assassins and Templars tend to have the most Evasion of all the classes because they rely on avoiding damage from melee attacks to stay alive.
A high Evasion attribute is important for any character who is hit by physical attacks, or spends time in melee range with foes. However, any class that receives physical damage can benefit from more Evasion.