The reason behind the definition of manifold I was going thorough the definition of a manifold and needless to say it wasn't something that I could digest at one go.
Then I saw the following Quora link and Qiaochu's illustrative answer.
It was great to see the motivation behind the concept of manifold. Then I looked at the definition of manifold that I have at my disposal which is the following:

A topological space $M$ is an $n$-dimensional real manifold if there is a family of subsets $U_\alpha$, $\alpha \in A$, of $\mathbb{R}^n$ and a quotient map $f \colon \coprod_\alpha U_\alpha \to M$ such that $f|_{U_\alpha}$ is a homeomorphism onto the image for all $\alpha$.

I understand that we are trying to conceptualize about a bigger space which when looked at a very small region looks like something else (a euclidean space) thereby giving a possibly incorrect bigger picture about the shape. Now what was the reason behind introducing disjoint union in this definition. Can someone help me to get in terms with this idea?
I went through this related question and Samuel's brilliant answer to it.
Leaving aside the points about Haussdorff and second countable spaces I could draw that homeomorphism is the concept that we use to convey the similarity between two spaces. I can loosely convince myself that the existence of homeomorphism between two spaces means the similarity in the pattern of open sets in the two spaces. (I might not be expressing what I feel about it.) But still then can anyone make it a bit more elaborate as to why we use homeomorphism here? If we divided this bigger surface, let's say earth into smaller circles, then I can see that we wouldn't have gotten the local shape same everywhere, somewhere it would have been a circle and some other points it would have been an area in between four circles or maybe something else. But are these local shapes homeomorphic to $\mathbb{R}^2$? What are other shapes which wouldn't have been homeomorphic to $\mathbb{R}^2$?
I guess "What happens when two spaces are homeomorphic?" would be a good way to start the discussion.
 A: "Being homeomorphic" is a very strong condition for spaces - it means that we have a bijection between the underlying sets of the spaces that is continuous, and whose inverse is continuous. In particular, you also get a bijection between the open sets of the two spaces.
We can therefore consider two spaces that are homeomorphic to be "essentially the same" for the purposes of topology.
So what goes wrong with your "Earth" analogy? I suspect when you say "circle" you mean "disc" (a circle is e.g. $\{x \in \mathbb R^2 \: | \: |x| = 1\}$ while a(n open) disc is e.g. $\{x \in \mathbb R^2 \: | \: |x| < 1\}$). Circles aren't open in the normal topology on a sphere, so we definitely want to be thinking about discs.
You say "some other points it would have been an area in between four [disks] or maybe something else" - this is true, but when dealing with a manifold we're always careful to make sure that the open disks we choose completely cover the manifold. This means that every point $p$ is contained in some $U_\alpha$.
I think part of the confusion is that the definition you've given isn't (to my mind) the most natural. In my opinion the nicest way to approach it is as follows: a manifold is a topological space $M$, and a collection of open sets $\{U_\alpha : \alpha \in A\}$ that cover $M$ (so given $p \in M$ we can find some $\alpha$ with $p \in U_\alpha$), and a corresponding collection $\{V_\alpha : \alpha \in A\}$ of open subsets of $\mathbb R^n$, and a collection of homeomorphisms $\phi_\alpha: U_\alpha \to V_\alpha$. (There are also the hypotheses about Hausdorff, second countable etc., but these aren't key for understanding the idea of a manifold.)
This definition is equivalent to the one you gave, but it makes it clearer that what we're caring about is that at every point $p$, the manifold $M$ locally looks like a subset of $\mathbb R^n$.
I'm not sure if that completely answers your questions, feel free to comment if there was something more specific that was confusing you.
A: The definition you've given is using a disjoint union as a sort of shorthand for a collection of maps. After all, one map $f: \coprod_\alpha U_\alpha \to M$ is equivalent to a collection of maps $\{f_\alpha: U_\alpha \to M\}$ where $f_\alpha = f|_{U_\alpha}$. The $f_\alpha$ are the charts for the manifold which identify an open subset of $M$ with an open subset of $\Bbb R^n$.
The definition of manifold which I'm more familiar with is the following: A topological space $M$ is an $n$-dimensional manifold if there exist open subsets $U_\alpha \subseteq \Bbb R^n$, $V_\alpha \subseteq M$, and homeomorphisms $f_\alpha: U_\alpha \to V_\alpha$ such that $\bigcup_\alpha V_\alpha = M$ (i.e. every point in $M$ is in some chart $V_\alpha$). We also assume that $M$ is second countable and Hausdorff. Personally, I find this definition to be a bit more transparent. It's easier to visualize what this definition is saying.
Now, for your question about what a homeomorphism really means. Intuitively, two spaces are homeomorphic if they are completely equivalent for the purposes of topology. Any topological construction you make, or topological property you prove about one space will hold for a homeomorphic space. They really are exacly the same.
More technically, a homeomorphism $f: X \to Y$ is firstly a bijection of sets, so it shows that if you start with $X$ and just relabel the points, then you get $Y$, as a set. More than that though, $f$ also induces a bijection on the sets of open subsets. Let $\tau_X$ be the topology on $X$, i.e. the set of all open subsets of $X$. Likewise for $\tau_Y$. Then we have a bijection $f: \tau_X \to \tau_Y$ given by $U \mapsto f(U)$. Therefore, the homeomorphism $f$ shows that if you start with $X$ and relabel its open subsets, you end up with the open subsets of $Y$. So if all you care about is the collection of open subsets of a space (i.e. you're doing topology), then homeomorphic spaces only differ by how you labeled the points and open subsets (which obviously doesn't make any tangible difference).
