Regen is short for regeneration and is the quantity at which you regen per cycle. With 0 Regen bonus, every 25 seconds you regenerate 2% of your character's Vita and Mana. This means if you were 100/50 stats and were injured, every 25 seconds you would regen 2k vita and 1k mana until fully healed.
Regen bonuses from items increase the percentage you regenerate each time. Everyone 1 point of Regen bonus is a 0.02% increase in the amount you will regenerate. So for instance, if you had a Regen bonus of 10 with the same stats, every 25 seconds you would regen 2.2% or 2200/1100 each time.
Regen bonuses can be very useful when it comes to soloing or PKing 1 on 1. Purple potions also boost your Regen bonus for a short period of time.