Intuitive understanding of manifolds

Intuitively, a manifold is some space such that if you zoom in enough, it looks like flat euclidean space. Let us call one of these small, flat patches a "chart" (so the chart is just what you can see when you've zoomed in sufficiently).

We need to be able to cover the entire space with such charts, and the space can't have crazy stuff happening where the charts overlap.

For example, the graph of the curve $y=x^2$ is a manifold because for any point on the graph, we can zoom in far enough so that the tangent line is a very good approximation.

On the other hand, the graph of $y=|x|$ is not a differentiable manifold, because no matter how far we zoom in to the point $(0,0)$, there is always this sharp edge. Note that this is a valid topological manifold, but not differentiable. (I sloppily read the question to mean differentiable manifolds.)

The topology comes in when you describe what types of sets the charts can be. Your charts must be equivalent (topologically) to an open set in $\mathbf{R}^n$ (Euclidean space). There are also some other technical conditions which rule out crazy pathologies that people can come up with.

Definition of a topological space is already given above. I will just try and explain my visualization of a manifold. As said earlier, its a (topological) space that locally looks like $ R^n $ . Some people also interpret it to mean locally flat. A very common example is the globe representation of the Earth (geographical). This encompasses the notions of "charts and atlases". A chart is nothing but the co-ordinate mapping as defined by Berci. And a collection of open sets in the space and the corresponding charts $ \{U_x ,\phi_x\} $ is called an atlas on the manifold. Note that there can also be possible overlapping of the domains of the charts.

This is a very crude analogy. You can just picture a manifold as a dark room with a torch available. So at any given time, the torch will permit you to look at a certain region of the room only hence giving you a local idea about how the room looks like. The torch here is nothing but $ R^n $ which comes equipped with a nice co-ordinate system.That the torch is in this room is what makes it a manifold.

For a topological space to be an "interesting" manifold, it has to satisfy certain topological properties like Hausdorffness and paracompactness, etc.

Books are aplenty. One would be J.M.Lee's Introduction to smooth manifolds. One book which I have tried a bit but enjoyed the little I have read of it is by Dubrovin, Novikov, Fomenko. I would also suggest notes uploaded on some reputed University websites by the professors there. They are very helpful for self study.

A $U\subseteq\mathbb X$ is an open set (for $X=\mathbb R^n$ or $X$ any metric space) if it is a union of regular open balls, i.e. whenever $x\in U$, there is an $\varepsilon>0$ such that $B_x(\varepsilon)\subseteq U$. Intuitively it means, that $U$ "doesn't contain its border".

A topological space is given by a set $X$ of its points and the class of its open sets, taking these as primitive, but the main point is that we get a notion of limit and continuity. The class of open sets must be closed under finite intersection and arbitrary union. Topological spaces $X$ and $Y$ are topologically the same (homeomorphic) if there is an $f:X\to Y$ bijection that is also a bijection between the open sets (i.e., both $f^{-1}$ and $f$ are continuous).

A topological/differentiable/smooth $n$-manifold is a topological space $X$, which is locally $\mathbb R^n$. Precisely, we require that for every point $x\in X$ there is an open $U\ni x$ and a homeomorphism $\phi_x:U\to B$ where $B$ is any ball in $\mathbb R^n$ ($B=B_z(r)$), such that all $\phi_y\circ\phi_x^{-1}$ maps are continuous/differentiable/smooth. These $\phi_x$ are also called the coordinate mappings.