Why are quadrants defined the way they are?
Solution 1:
Ultimately it's just a convention, and it could equally well have been different. But there are some other conventions it ought to align with (so they would possibly also have to change to make everything fit together):
As Yagna Patel points out, quadrant numbering is intimately connected with the standard parameterization of the unit circle. Change one and you'll have to change the other to fit (if you don't want to go crazy).
In the complex plane, the standard parameterization of the unit circle ought to align with $t\mapsto e^{it}$, so once we have decided how to parameterize the circle, we get the position and orientation of both the real and imaginary axis.
Going back to two-dimensional coordinate geometry, we'll want to parameterize the unit circle such that $t=0$ is at the intersection of one of the axes. And it's also somewhat easier to think about things if a small positive $t$ corresponds to the quadrant where both coordinates are positive, so that has to be the first quadrant.
The horizontal axis could be either the first or second coordinate in a coordinate pair -- but it should grow from left to right because that is the reading direction in the language the trendsetting mathematicians communicate in.
Finally, we can number the quadrants either 1,2,3,4 or 0,1,2,3 without much influence on other conventions.
How these considerations work out is that as long as our basic coordinate system has the axes growing towards the right and up, there are two orderings of the quadrants that make things fit together, namely NE NW SW SE and NE SE SW NW. The first of this is the convention pure mathematics uses, the second would align with the scale for compass directions used in practical (air/sea) navigation. (It's a minor nuisance that these conventions are different, but neither is realistically changeable).
If we also consider systems where the vertical axis grows downwards (which is quite common in computer graphics, ultimately because that's the direction line numbers grow in in text), we get the possibilities SE SW NW NE and SE NE NW SW.
Solution 2:
My understanding is that the quadrants are numbered in the order an angle in standard position passes through them as the angle increases - that is, counterclockwise starting at the x axis. Additionally, because that's the direction that the point $(\cos x, \sin x)$ goes when $x$ goes from $0 \text{ to } 2\pi$ (it makes a circle).