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The definition is the same as in Mathematics and, I suppose, elsewhere:

it is a topological space such that the vicinity of each point is homeomorphic to a disk in $\mathbb{R}^n$ (note, that dimension has to be the same for all points $x$). This requirement is important, since not every imaginable geometric object satisfies this requirement:

  • Sphere is a manifold, since one can draw a tangent plane in the vicinity of any point. It is "locally flat" and even we humans see Earth flat, since its radius of curvature is much greater than the visible distance (let the proponents of the flat Earth theory forgive me).

enter image description here

  • Two intersecting lines are not a manifold, since for any point, except for the intersection we have a 1-dimensional space and for the intersection point, there are two non-collinear vectors belonging to them.

enter image description here

Natural examples emerging in Machine Learning are images, videos, or arbitrary data. One usually treats, say, an image, as an object in the $\mathbb{R}^{H \times W \times 3}$, where $H$ is the height, $W$ - width of the image, and $3$ - number of colors. But in fact, only a small subset of all objects in this high-dimensional space are real images, and they belong to some manifold of a lower dimension.

It is a non-trivial question to tell what exactly the true dimensionality of data is. For MNIST, it is claimed that it is $3$ (instead of $28 \times 28 = 784$).

As a good material on this topic, I recommend this lecture from the recent workshop.

Scikit-learn has a nice exposition as well.