Matrix Norm

Dear All,

I get confused of this definition:

||A|| = sup ||Ax||/||x||

can anyone explain about it?
what is it mean? and also where that come from?
because as far I know
||A|| = (x1^2 + x2^2 + ... + xn^2)^1/2

I really appreciate if you willing to answer my question.

Thanks and regards,
Henry
.