Purifying your gear can give 4 additional attributes, these attributes depend on your gear.
Helmet, Clothes, Shoes, Talisman and Artifact can gain elemental (Fire, Holy, Thunder, Ice, Darkness, Poison) defence, crit tough or crit resistance.
Weapons and Rings can gain elemental attack, crit damage or crit hit.
When purifying, you can lock a desired attribute by tapping the lock icon. This means that you'll always keep the locked attribute. However, this doesn't mean the figure on the attribute won't change (It can increase or decrease, unless you use blue diamonds to purify).
If you don't lock any attributes, it costs 1 purification stone. If you have attribute locked, it costs 2. If you have 2 attributes locked, it costs 4. If you have 3 attributes locked, it costs 8. And finally, if you have all attributes locked, it costs 16. You can also purify using jewels and blue diamonds.
The colours of the results indicate the numerical differences to the maximum values:
Inheritance is a mean to pass on your purification attributes to your new gear. The gear you're inheriting to must be the same or higher level than the gear you're inheriting from. The amount of purification stones needed for inheritance varies, but if you've purified your old gear a lot, it's definitely worth inheriting.