The order of the generator point G used in bitcoin
It's easy to verify the order (n):  Multiply G by n and find that you get the point at infinity. This proves that n is either the order or a multiple of it. Then convince yourself n is prime using a Baillie-PSW primality test, so it must be the order itself and not a multiple of it.

Finding the order is not quite so simple as verifying it. To do so you would use Schoof's_algorithm, which requires quite a bit of number-theory complexity. The Wikipedia article on Schoof's explains how Hasse's theorem is critical to this process by restricting the range of orders that could possibly be the correct one.