Having trouble with (simplicial) homology

Well, I've been to over two weeks' worth of lectures and reading the relevant chapters in Hatcher, but I still have no idea how homology works.

Here's what I understand

you have n-simplices like points, lines, triangles, tetrahedrons, etc.

then you have boundary maps, which take the simplex and their rotation and give the faces. For example if you had a triangle with vertices x,y,z then the boundary map of this simplex gives [x,y]-[y,z]+[z,x]...

Ok. So [x,y] would be the 1-simplex between 0 and 1. What does the sum [x,y]-[y,z]+[z,x]

The first example in the chapter is computing the homology groups of the circle. Okay, so a circle is one vertex v and one edge e.

The explanation given in Hatcher includes:

"Then and are both **Z** and the boundary map is zero since ."

I'm lost. This is supposed to be the easiest example, and I have no clue what he's rambling on about. Could someone help?