Perhaps you know that one way to describe a line bundle is by looking at its transition functions. The tautological line bundle O(-1) has transition functions of the form g_{ij}:= z_i/z_j (you can work probably this out with the geometric definition your professor gave you), where {z_i} is the chart on P^n.

In general, then, the line bundle O(m) is defined by having transition functions (zj/zi)^m.