Trivial power of line bundle I'm trying to understand the following:

Thinking of a line bundle as a bunch of locally generating sections together with transition functions (which in this case are just multiplication by local sections of the structure sheaf) I wonder what it means that some power of a line bundle is trivial. I guess an example with $N>1$ would answer my question.
Thank you.

This is meant as a complement to Remy's nice answer.
The first topological example is the Mobius bundle $L$ on $S^1$ - the transition functions can be taken as multiplcation by $1$ and $-1$. Squaring a line bundle squares the transition maps, so $L^{\otimes 2}$ is trivial. I think it's possible to make an algebraic version of this example by following the lead of example (4) on page 22 of this book.
There are also algebraic examples in 6.5.1 of Hartshorne: if you take $U = \mathbb P^n \setminus C$ where $C$ is the zero set of a degree $d$ homogeneous polynomial, then the line bundle $L$ on $U$ corresponding to the hyperplane divisor of $C$ will have $L^{\otimes d}$ trivial.
For example, in $\mathbb P^2$ let $C = V(x^3 + y^3 + z^3)$ and take the line bundle $L$ on $U$ corresponding to the hyperplane $x=0$, say. Then $L^{\otimes 3}$ will be trivial essentially because of the existence of the globally defined regular function $\frac{x^3}{x^3+y^3+z^3}$, whereas $L^{\otimes 2}$ will be non-trivial (there is no globally defined regular function of the form $\frac{x^2}{p}$).
Also, in this question it was noted that for normal projective varieties over $\mathbb C$ there exists a torsion line bundle if and only if $H_1(X,\mathbb Z)$ is non-zero, so they are quite common. 
