Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any open $U$ and $\widetilde{\mathcal M}$ be the presheaf given by $\widetilde M_U$ over an open $U$. Sheafification $\mathcal M$ of $\widetilde{\mathcal M}$ is called the sheaf of meromorphic functions on $X$. Its sections over an open $U$ will be denoted by $\mathcal M_U$. Then $\mathcal O_U$ is naturally included in $\mathcal M_U$. Now let $\mathcal O^\times$ and $\mathcal M^\times$ be the sheaves of multiplicative groups of invertible elements of $\mathcal O$ and $\mathcal M$. Again, $\mathcal O_U^\times \subset \mathcal M_U^\times$. We can consider the sheaf $\mathcal M^\times / \mathcal O^\times$. Its global sections are called Cartier divisors.
If I'm not mistaken, any section of $\mathcal M$ over an open $U$ is given by an open cover $\{U_i\}$ of $U$ together with a family of elements $\{f_i / g_i \mid f_i, g_i \in \mathcal O_{U_i}$, $g_i \neq 0 \}$ and such that $$ f_i|_{U_i \cap U_j} g_j|_{U_i \cap U_j} - g_i|_{U_i \cap U_j} f_j|_{U_i \cap U_j} = 0 $$ if $U_i \cap U_j \neq \varnothing$.Thus the inclusion $\mathcal O_U \to \mathcal M_U$ for any $f \in \mathcal O_U$ is given by the data of the single set $U_1 = U$ and of a single fraction with $f_1 = f$, $g_1 = 1_{\mathcal O_U}$.
Then any section of $\mathcal M^\times / \mathcal O^\times$ over an open $U$ is given by an open cover $\{U_i\}$ of $U$ together with a family of elements $\{f_i / g_i \cdot \mathcal O_U^\times|_{U_i} \mid f_i, g_i \in \mathcal O_{U_i}$, $f_i, g_i \neq 0 \}$ each of which represents a set and such that $$ f_i|_{U_i \cap U_j}/ g_i|_{U_i \cap U_j} \cdot \mathcal O^\times_U |_{U_i \cap U_j} = f_j|_{U_i \cap U_j} / g_j|_{U_i \cap U_j} \cdot O^\times_U |_{U_i \cap U_j} \tag{1} $$ if $U_i \cap U_j \neq \varnothing$. Are these considerations correct?
Yes, the main idea of a Cartier divisor is that you are locally describing a "codimension 1 subvariety with multiplicities" by giving rational functions on an open cover, with a compatibility condition saying that on the overlap of two open sets in the open cover, the two corresponding rational functions agree up to multiplication by a nowhere zero, everywhere regular function. Thus, the corresponding rational functions can be thought of as cutting out a single locus of zeros and poles (with multiplicities). The data of the Cartier divisor gives you a local equation cutting out this locus in each open set in your cover.
You must also be careful to keep in mind that two different collections $(\mathcal{U_i}, f_i / g_i)$ and $(\mathcal{V_j}, f'_j / g'_j)$ will define the same Cartier divisor if $f_i / g_i$ and $f'_j / g'_j$ restrict to the same rational function on $U_i \cap V_j$ (provided it is non-empty) up to multiplication by a nowhere zero, every regular function on $U_i \cap V_j$.