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Recall that a function $f$ from a set $A$ to a set $B$ exactly one element of $B$ to each element of $A$. The graph of f is the set of ordered pairs $(a, b)$ such that $b=f(a)$. Because the graph of $f$ is a subset of $A\times B$, it is a relation from $A$ to $B$. Moreover, the graph of a function has the property that every element of $A$ is the first element of exactly one ordered pair of the graph.

In this statement, what does $A \times B$ mean? does it mean the cartesian product of $A$ and $B$?

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    Yes, it is the cartesian product. – 2017-01-21

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Absolutely yes, it means the cartesian product.

The graph of a function is a subset of the cartesian product $A\times B$, with the few properties you have listed in your question.