When you make a successful Reflex save to avoid damage, you suffer no damage instead of half damage.
Feat synergy
Shadowdancer's fourth Core ability improves your Evasion by one step. If a character naturally and permanently possesses Evasion as a feat, you temporarily gain Improved Evasion.
If you additionally possess Improved Evasion as a permanent feat, you no longer automatically fail Reflex saves on a roll of 1.
Evasion and Liquid Luck are equivalent feats. Possessing either will allow you to obtain Improved Evasion. Possessing both will not grant any further benefit.
Notes
Evasion primarily helps against certain magic attacks as well as avoiding most types of trap damage. (The Spell wards in The Underdark adventures must be met by your Will save and cannot be evaded.) For avoiding most physical and ranged attacks, Dodge bonus chances are used.
There is no warning that you have lost evasion until you try to use it and cannot. Then, a text message appears at the top of the screen: "EVASION FAILED. You cannot evade <reason>"