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The product topology on a Cartesian product of topological spaces is generated by the basis of boxes (products of open sets from each factor). This topology makes coordinate projections continuous and is the coarsest topology with this property. It allows us to study properties of finite (and infinite) products of spaces.
Given two topological spaces (X, τ_X) and (Y, τ_Y), you want to put a topology on their Cartesian product X × Y. The natural candidate for open sets is to take products of open sets: U × V where U is open in X and V is open in Y. But you cannot take these as the open sets directly, because the collection of all such products is generally not closed under finite intersections in the required way (intersections of two such products are still products, so that is fine, but finite unions of products are not always products). Instead, these products form a basis: the open sets of the product topology are arbitrary unions of such basis elements. From your study of bases for topologies, you know this is the standard construction.
The key property that determines which topology to use is the universal property: the product topology is the coarsest (fewest open sets) topology on X × Y that makes the two coordinate projection maps π₁: X × Y → X (sending (x, y) ↦ x) and π₂: X × Y → Y (sending (x, y) ↦ y) continuous. "Coarsest" means we add open sets only as needed to achieve continuity of projections — no extra structure. This universal property characterizes the product topology uniquely and generalizes cleanly to any number of factors.
The simplest case illuminates the geometry. With ℝ × ℝ = ℝ² and the standard topology on each copy of ℝ (open intervals as a basis), the basis for the product topology consists of all open rectangles (a, b) × (c, d). The resulting topology on ℝ² is exactly the standard Euclidean topology — the one generated by open disks. Open rectangles and open disks generate the same topology because each contains the other: every open disk contains an open rectangle around each of its points, and every open rectangle contains an open disk around each of its points.
For infinite products ∏_{α ∈ A} X_α, two choices appear: the box topology (arbitrary products of open sets) and the product topology (products of open sets with only finitely many factors restricted). The product topology is the coarser one — it is the one that makes all projections continuous — and it has dramatically better properties. The Tychonoff theorem (an arbitrary product of compact spaces is compact) holds for the product topology but fails for the box topology. The box topology is too fine: it has so many open sets that compactness breaks down. This is why the product topology, not the box topology, is the standard definition, even though the box topology might seem more natural at first.