The D&D equivalent of evasion is AC (Armor Class). It's your imaginary way of saying "my armor deflected that attack/my armor stopped the attack/my natural evasion was enough that I completely avoided the attack" and such. An armored opponent gets AC from armor, the higher quality, the higher the AC. A monk gets a natural bonus to AC as he levels up, which get pretty high, usable as long as he doesn't wear armor. You also have spell resistance and Reflex, Fortitude, and Will saves. Different situations call for different saves.
Here's a character sheet (or at least one of a few pages)-
Character Sheet
In case people are thinking it, your class isn't limited to what is listed on the main page. You can invent one and post as long as it's balanced. The balance doesn't come from D&D, it comes from you not saying, "My class is the super rape knight of awesomeness. It destroys everyone else here and I win."
It could be something like a guy that specializes in using improvised weapons, has a knack for saving babies, and always gets the girl.