In a category $C$, a product of two objects $X$ and $Y$ is an object $X \times Y$ and two projections $\pi_1\colon X\times Y \to X$ and $\pi_2\colon X\times Y \to Y$ satisfying the following universal property: If $A$ is some object and $f\colon A \to X$ and $g\colon A \to Y$ then there is a unique morphism $(f,g)\colon A \to X\times Y$ satisfying that $\pi_1(f,g) = f$ and $\pi_2(f,g) = g$. To get intuition for this consider the product in set and check that it accords with this definition.