A manifold is a space that looks flat and ordinary up close, even if it is curved or strangely shaped overall. The surface of the Earth is the classic example: stand anywhere on it and your immediate surroundings look like a flat plane, even though the planet as a whole is a sphere.
Spacetime is treated as a four-dimensional manifold in general relativity. Zoom in close enough to any single point and the rules of ordinary flat geometry apply, but stitch enough of these local patches together and the whole thing can be curved by the presence of mass and energy.
This local-flat, global-curved structure is what lets physicists do calculus on spacetime at all. Every point has a small flat neighbourhood where the familiar mathematical tools work, and the curvature only shows up once you compare how those neighbourhoods fit together.