In Remark 5.1.3. of her book Hochschild Cohomology for Algebras Witherspoon writes (I am partly summarizing), for $k$ a field:
Let $R$ be a commutative $k$-algebra together with an augmentation $\epsilon\colon R\rightarrow k$. Assume that $R$ is $\operatorname{ker}(\epsilon)$-adically complete. A deformation $A$ over $R$ is an associative $R$-algebra $A_R$ that is isomorphic to the completed tensor product of $A$ with $R$ as an $R$-module and for which there is a $k$-algebra isomorphism $A_R/\operatorname{ker}(\epsilon)\xrightarrow{\sim} A$.
What is "the completed tensor product of $A$ with $R$ as an $R$-module" in this context? Is the above definition of an (algebraic) deformation developed somewhere in the literature? In Witherspoon's book it remains only a side remark and is not developed further.