What is $S^2 \times S^2$? What does the product of these spaces mean? 
I really cannot understand or wrap my head around it.
What is it done for?
If someone could help in visualizing it or provide intuition, it would be great.
 A: As @Lee Mosher pointed out, the definition of the cartesian product,
$$ X \times Y = \{(x,y): x \in X, y \in Y\}$$
Well, since we are doing topology, a more geometric definition would be beneficial. Hence, you can think of the product in two following ways,
1) At each point $x$, there is a copy of $Y$
2) At each point $y$, there is a copy of $X$
Why does this help? Well think of $\mathbb{R} \times \mathbb{R}$. If we think of this as joining $ \cup_x\{x\} \times \mathbb{R}$ in the plane, then we see a bunch of copies of lines i.e the plane is built from lines either in the horizontal or vertical direction. This is where you can see the idea of fibers/bundles extending from. The plane is therefore realized as a line bundle over $\mathbb{R}$ with fibers $X$ or $Y$, depending on how you choose to look at things. 
$\textbf{Addition}$: Also, cartesian products of manifolds is the most straightforward way of getting a new manifold. You have to think a bit when it comes to getting new manifolds via quotients e.g $\mathbb{R}/ \mathbb{Z} \cong S^1$. Old manifold $\mathbb{R}$ and we get the new one $S^1$ by defining an equivalence relation. 
I hope this helps. 
A: Product spaces are very easy to understand. They are spaces consisting of multiple other spaces, in each of which you may choose a point.
For instance, in this example, we have two spheres $S^2$. A point in this product space is a selection of a point on each sphere.
For an intuition: maybe we want to plant a flag on Mars and the Moon. Where on the surfaces shall we plant these flags? We have two spheres, on each of which we may choose a point on the surface. This space of choices is $S^2 \times S^2$.

A: My advisor would draw a rectangle on the blackboard and label it $E$, a horizontal line parallel to the bottom edge and label it $M$, and say, "Let $p:E \to M$ be a principal bundle."
In that spirit (but embellished for illustration), $S^{2} \times S^{2}$ looks like this:

Each horizontal line in the grid represents a $2$-sphere that projects to a single point of the left-hand sphere. Each vertical line represents a $2$-sphere that projects to a point of the bottom sphere.
If you stare long enough (perhaps over a period of months or years) at diagrams of this type, you develop a degree of "geometric intuition" that may include being able to see (in roughly increasing order of familiarity):


*

*How two of the $2$-spheres described above are either disjoint or intersect in one point;

*How ordinary flat $2$-tori are contained in the product;

*How $S^{2} \times S^{2}$ is obtained from the Cartesian product of two $2$-disks (whose boundary comprises two solid tori intersecting along a $2$-torus "corner") by pinching the corner to a single point while "zipping up" each solid torus by collapsing its circle factor;

*How to obtain this closed $4$-manifold from the complex projective plane through birational transformations (blowing up two points $p$ and $q$, then blowing down the proper transform of the line through $p$ and $q$);
and so forth.
