How do games generally make enemies move in really curvy ways like in Blazing Lazers?

If they are on a set pattern, trigonometric functions are useful. If their movement is not on a set course, such as if they are reacting to the player, then there are a few methods to achieve those smooth curves. The main way is to model their movement as vectors and change an object's movement using velocity and accelerating vectors.
eg: If an object's velocity is (3,0) and its acceleration is (-1,2) for 3 seconds, then it will smoothly change course from east to north while accelerating. In the case of a game like blazing lazers, this is like simulating thrust.