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One of the most important results in operator theory is And\^o's \cite{ando}generalization of dilation theory for a single contraction to a pair ofcommuting contractions acting on a Hilbert space. While there are two explicitconstructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of theminimal isometric dilation of a single contraction, there was no such explicitconstruction of an And\^o dilation for a commuting pair $(T_1,T_2)$ ofcontractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. Inthis paper, we give two new proofs of And\^o's dilation theorem by giving bothSch\"affer-type and Douglas-type explicit constructions of an And\^o dilationwith function-theoretic interpretation, for the general case. The results, inparticular, give a complete description of all possible factorizations of agiven contraction $T$ into the product of two commuting contractions. Unlikethe one-variable case, two minimal And\^o dilations need not be unitarilyequivalent. However, we show that the compressions of the two And\^o dilationsconstructed in this paper to the minimal dilation spaces of the contraction$T_1T_2$, are unitarily equivalent.  In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to0$ strongly, an And\^o dilation was constructed recently in \cite{D-S-S},which, as this paper will show, is a corollary to the Douglas-typeconstruction.  We define a notion of characteristic triple for a pair of commutingcontractions and a notion of coincidence for such triples. We prove that twopairs of commuting contractions with their products being pure contractions areunitarily equivalent if and only if their characteristic triples coincide. Wealso characterize triples which qualify as the characteristic triple for somepair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a purecontraction.

And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models