You're asking a very big question. So, I'll just give a quick over view.

Normal backgrounds are like maps that can be shifted horizontally or vertically on the screen. The one background in Mode7 however can do more than just shifting, you can scale/rotate/warp it however you want with matrix operations.

How does the PPU know which pixel to display where? It puts the (x1,y1) coordinates of the pixel on the screen into the equation below. Which tells it what pixel (x2,y2) to take from the 'BG map'.

There are registers to allow you to set all the values in this matrix equation (a,b,c,d,x0,y0).

from the nintendo documentation:

		/ x2 \   / a b \ / x1-x0 \   / x0 \
		|    | = |     | |       | + |    |
		\ y2 /   \ c d / \ y1-y0 /   \ y0 /

		x0&y0: center coordinate
		x1&y1: display coordinate
		x2&y2: coordinate before calculation

an example matrix transformation from nintendo:
	to scale and rotate the picture -       

	a=cos(gamma)*(1/alpha), b=sin(gamma)*(1/alpha)
	c=-sin(gamma)*(1/beta), d=cos(gamma)*(1/beta)

  	gamma: rotation angle
  	alpha: reduction rates for x(h)
  	beta : reduction rates for y(v)