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.
Manifolds show up constantly outside physics too. The surface of a globe is a two dimensional manifold, and it is exactly why flat maps always distort something, Greenland looks enormous on a standard Mercator projection because no single flat coordinate system can represent a curved sphere without some kind of stretching. Coordinates on a manifold can also simply break down at certain points, longitude lines all converge and become meaningless exactly at the north and south poles, even though nothing physically strange is actually happening there. Physicists run into an analogous issue with certain coordinate systems used to describe black holes, where the mathematics appears to blow up at the event horizon even though, with a better choice of coordinates, nothing physically catastrophic occurs at that boundary at all.
The mathematical framework for manifolds was developed by Bernhard Riemann in the 1850s, decades before Einstein needed it, originally as pure mathematics with no physics application in mind. Einstein spent years learning this unfamiliar branch of geometry, with help from mathematician friend Marcel Grossmann, before he could finally express general relativity in a form built on manifolds rather than flat, Newtonian space.