Riemannian geometry is the geometry of objects that when you look at small pieces of them, they look like Euclidean space.
For example, the surface of a sphere. If you look at a small neighborhood of any point of a sphere, it looks like flat space. For example, right now you are on a sphere and the earth around you looks like flat space.
Another example is a donut. Locally, it looks like flat space, even though in total, it is clearly not flat.
Riemannian geometry is very powerful and useful field of mathematics, and usually falls under the broader umbrella of Differential Geometry.