What makes a hydrocarbon saturated?

Explanation:

It's easier to answer what makes it unsaturated.

H - C ≡ C - H

This is Acetylene. It has three shared bonds between the two carbons. That many bonds makes it unsaturated.

     H   H

H - C - C - H

     H   H

Draw lines from 1 H on top to the Carbon underneath Do the same with the other hydrogen. Make the two Hydrogens on the bottom do the same thing.

The point is that Carbon has 4 bonds. 1 bond goes to the other Carbon. The other 3 take on Hydrogen. That molecule is saturated: nothing can be added in its natural state.