Torus construction I understand the usual way to construct a torus. i.e., pasting opposite 
edges of a rectangle.
But I don't know the construction by saying take a zero cell and attach
a two one cell and then attaching a two cell.
Also construction of a orientable surfaces of genus 2g from a polygon with
4g sides.
I am not able to imagine. Could you please help me to imagine this?
Thanks in advance.
 A: Fortunately, the one you understand can be readily seen in the cell-complex construction.
So, take a rectangle, identify opposite sides. Now, draw a picture of a torus, and draw the rectangle on it. This is very important that you can do this. What does the rectangle look like on the torus? It likes like a sort of figure-8 (sort of). All four corners are the same point. Two pairs of opposite sides are associated, so we get only 2 edges, not 4. Good.
This is how the cell-structure comes, too. Take one point (0-cell). Take 2 1-cells (each loop in the figure 8). And take 1 2-cell. But how do we attach our 2 cell? Well, a 2-cell is just a square. So on the original rectangle that you drew and understood, why don't you just take that to be your 2-cell? Then the attaching maps are precisely those implied by your drawing.
So the cell-structure and the rectangle are, in fact, the exact same.  In fact, when I give cell structures for genus-g surfaces, I give them in that fashion.
It all comes down to (in my opinion) finding that figure 8 on the torus itself, to understand what that rectangle is. If this doesn't make sense, comment, and I'll upload an image.

Here, we see the figure 8 and the two loops. All four corners are the same point. Their intersection is the 0-cell, the red and the blue are each 1-cells, and the surface is a single 2-cell is attached with the following attaching map (where a is the red side, b is the blue)

A: Take a sheet of paper, and curl the far end over to make a cylinder.  The place where the edges of the paper meet is one of your one cells.  Now fold the ends of the cylinder over to touch each other and make a torus.  The circle where these ends meet is another one cell.  The place where your circles intersect is a 0 cell.  Clearly, the paper itself is the two cell.
Its not really all that different from the pasting-opposite-edges-of-rectangle approach.
