Understanding the properties and use of the Laplacian matrix (and its norm)

I am reading the wikipedia article on the Laplacian matrix:

http://en.wikipedia.org/wiki/Laplacian_matrix

I don't understand what is the particular use of it; having the diagonals as the degree and why the negative adjacency elements off the diagonal? What use would this have?

Then on reading about its norm, first of all what does a norm really mean? And what is the norm for the Laplacian matrix delivering? This norm does not result in a matrix whose terms cancel out or sum to one. Or that the determinant is equal to any consistent value. Any insight?

Best,