Intuitive understanding of manifolds I'm new to topology, but I must understand how it works to progress in my research. 

First, can anybody point me to a document that introduces topology in a "gentle" way? What pre-requisites do I need to understand the topic?

I need an intuitive understanding of manifolds for now.
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.
I guess these are all good explanations. I would like to give my idea of a manifold.

Suppose you have a geometrical object and you can relate it or give a one-one correspondence with any Euclidean space at every point, then this geometrical object is called manifold. Simple examples are lines, circles, planes, and torus.