A manifold is a mathematical space that, on a small scale (locally), resembles a Euclidean space.
It may help to visualize a crumpled piece of paper and observe that we can zoom in enough to where we may a flat plane. This would be a 2D space embedded in a 3D space.
Another fun example is flat-earth theory: how do flat earther’s perceive that the Earth is flat when in fact it is a large sphere? This is because that on a small-scale the earth really appears flat! In general, any object that is nearly flat on a small scale is a manifold.
Formal Definition
A manifold is a topological space that is locally homeomorphic to a Euclidean space. This means that each point on the manifold has a neighborhood (a small region around the point) that can be continuously mapped to a Euclidean space without tearing or gluing.
Why?
With manifolds, it is possible to capture intrinsic structure of high-dimensional data within lower-dimensional spaces. There are computational motivations for pursuing such a feat, see: Dimensionality Reduction