What is manifold in Geometry?

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand.

A manifold is a topological space that is locally Euclidean

can anyone explain it to me please

There are different kind of manifolds. It's one is being used for specific theories.

For example in topology (like algebraic topology) we need the space to be Hausdorff, second-countable and locally Euclidean in order to be a topological manifold. There you can build charts from the manifold to the Euclidean space via homeomorphisms and eventually get an atlas in your manifold. For example if $M$ is a top.manifold and for every $p\in M$ there is an open neighbourhood of $p$ say $U$ and a homeomorphism $\phi:U\to \phi(U)\subset \mathbb R^n$ with $\phi(U)$ open.

What you want is get info about an area of your manifold. So you "throw" this area in $\mathbb R^n$ where you know how the things work there and then you get the info back with $\phi^{-1}$.

Also if you allow more restrictions in the definition of the manifold like that we want diffeomorphisms in the place of homeomorphisms you get differential or smooth manifolds.

The $S^2,\mathbb RP^2$ are top. manifolds (and diff. manifolds but this is another story,you need to define much more things) in $\mathbb R^3$(embedded) of dimension $2$ where $S^1$ is of dimension $1$.

If you need more strict(mathematical) details tell me.

Consider a two-dimensional manifold embedded in three-dimensional space. Such manifold would look locally like a sheet of rubber: it might be dirtorted, bent, but it must not be cut. It must not have a border, so it might either be infinite or bend back onto itself. You can't have it tapering to a one-dimensional string, because it has to look two-dimensional everywhere.

But the surface doesn't have to be embedded into some higher-dimensional space, or it might do so only with self-intersection. So although visualizing manifolds as embedded, you should try to assume the viewpoint of some insect living on that surface, taking note of your two-dimensional surroundings but not any properties of the ambient space.

This might give you some intuiton; if you need stricter definitions, you'll likely have to cope with the definitions given in literature, e.g. Wikipedia.