Rigidity Theorem, meaning. What does it mean to say that a theorem is a rigidity theorem? I'm reading a book (Lipman Bers, a Life in Mathematics) that says for example that Torelli's theorem is a rigidity theorem. So what does this mean: Rigidity Theorem? thank you.

What you want is the reason for naming a theorem this way. So the answer has to be some kind of guess and by analogy.

It is possible in algebraic topology to find two non-homeomorphic topological spaces having the same fundamental groups (or homology groups). That is within a "collection of topological spaces" of same fundamental groups one can find two different spaces.
In that collection one can push the given object to a nearby object which is different. (so it is not rigid).

Torelli theorem means that it is not possible to find similar behaviour for algebraic curves with same Jacobian.

Correct way to formulate rigidity would be to using the concept of parametrised family. There is a theorem in Humphreys Linear Algebraic Groups on Rigidity for Torus (tori). That will give more precise understanding of this.