There's a small error in the formula for O(N): the way he's written it it looks like for all n, there is a k such that kg(n) >= f(n), ie k depends on n so take k = f(n)/g(n) and all nonzero functions trivially satisfy it. It should be there exists a k such that for all n kg(n) >= f(n). Pedantic I know, but on the other hand I wouldn't call these "beautiful equations" associated with O(N), I'd instead call them the definition of O(N).
There's also o(f(n)), g(n) is a member of o(f(n)) if the limit as n goes to infinity of g(n)/f(n) is zero. Finally, there is asymptotic equality: f(n) ~ g(n) if the limit as n goes to infinite of g(n)/f(n) = 1. O,o and ~ are all subtly different, but if you're just trying to prove upper bounds then O(f(n)) is the one that comes up most frequently, which is why it's probably the only sort of asymptotic analysis most CS grads know.