Can you say something about the audience of this course? Popular math? Undergrads? grads? That might set some appropriate response parameters.
If this were a graduate-level course (I suspect not, but I feel like addressing this option anyway :)), I'd probably point to categories, sheaves, and cohomology -- and maybe just "cohomology" as a general concept, if I had to pick one. Also, the link provided by algebraic geometry between manifolds, varieties, and commutative algebra.
For an undergraduate non-major course, I don't think there's any way of overstating the historical significance of calculus. The scope of problems, both mathematical and physical, that were instantaneously solvable by mathematicians all over the world after its development and deployment, was mind-boggling.
I think there be are probably more important ideas than those above if the scope of the question is how much impact they've had on humanities' development (e.g., development of serious linear algebra would certainly go in there for applications to just about everything)everything, someone else mentioned RSA), but those the above are my votes for ideas that have changed the way that people (or at least mathematicians) have thought about mathematics.
