What are Dynamic Shadows?

Dynamic shadows are shadows that are calculated in real-time to look like the object's real shadow would, based on the shape of the object and the location of the light-source(s).

This is a computationally-expensive task, so it was not feasible in most older 3D games. Instead, they'd precompute shadows for static (non-moving) objects, and emulate shadows for moving objects, typically using a decal painted onto the ground.

Since you mentioned Watch Dogs, here is a comparison of Watch Dogs with and without dynamic shadows (please note this is an unofficial mod, so the shadows look weird sometimes):