Let $X$ be a scheme. $\mathcal K $ be the sheaf of total quotients of $X$. $\mathcal K^*$ be the sheaf corresponding sheaf of invertible elements of $\mathcal K$. So $\mathcal O_X^*$ is a subsheaf $\mathcal K^*$.
An Cartier Divisor on $X$ is by definition a global section of the sheaf ${\mathcal K^*} / {\mathcal O_X^*}$.
Now I want to prove that giving an global section of ${\mathcal K^*} / {\mathcal O_X^*}$ is same as providing the following data: $\mathrm{collection}{(U_i,f_i)}$ such that $f_i \in \Gamma (U_i, \mathcal K^*)$, where ${U_i}'s$ covers X and $f_i/f_j \in \Gamma(U_i \cap U_j, \mathcal O_X^*)$.