Mass is said to create curvatures in space-time thereby creating gravity, yet technically the smallest movements, even on Earth, create gravitational waves. Are there different "types" of disturbances of space-time? How?
Spacetime curvature is actually caused not by mass but by the density and flow of energy and momentum. These quantities are encapsulated in something called the “energy-momentum-stress tensor”. Even massless particles like photons can cause spacetime to curve because they have energy and momentum even though they don’t have mass. But this is a tiny theoretical effect, and of course realistically it is the energy of massive particles that causes gravity.
So I’ll answer your other question by talking in terms of mass. If a distribution of mass is sitting still, it creates a static gravitational field that decreases at large distances like $1/r^2$. But if it is moving in a certain way, it radiates a gravitational wave whose field decreases like $1/r$. In order to radiate, the quadrupole moment of the mass distribution must have a nonzero third time derivative. (Or the octupole moment can have a nonzero fourth derivative, etc.) This means, for example, that a spinning sphere does not radiate, but a dumbbell spinning end over end does.