Approval is a formal agreement. It's probably not a good idea to mail the invitations for your beach party until you've gotten official approval for the enormous bonfire you're planning to light.
Building designs, budgets, grant applications — these all require official approval before they can be put into effect. You can also use the word approval to talk about less formal permission, like a teacher's approval of your plan to buy pizza for the whole class. Approval also means the belief that something is good, like your approval of your neighbor's purple house. Until 1800, it was much more common for people to use the word approvance to mean the same thing.