Why is a field conservative if its irrotational, and why is this equivalent to saying it is described by some potential function?

Question

We were studying conservative fields today and I am a bit confused as to how the condition $ \nabla \times \vec F = 0 $, the field is irrotational, is equivalent to saying it is described by some function $f$ such that $F=-\nabla f$. Why is this the case. I dont see why the first reason is true, and much less why they are equivalent. I also don't see why the second case isn't true for all functions? If you integrate $F$, surely you can always find it gives a function $f$?