Generally, if $\rm\,R \subset S\,$ are rings and $\rm\,s\in S\,$ then $\rm\,R[s]\,$ denotes the ring-adjunction of $\rm\,s\,$ to $\rm\,R,\,$ i.e. the smallest subring of $\rm\,S\,$ containing both $\rm\,R\,$ and $\rm\,s\,.\,$ Equivalently $\,\rm R[s]$ is the image of $\rm\,R[x]\,$ under the evaluation map $\rm\,x\mapsto s,\,$ i.e. elements of $\rm\,S\,$ writable as a polynomial in $\rm\,s\,$ with coefficents in $\rm\,R.\,$
The notation for the polynomial ring $\rm\,R[x]\,$ is the special case where $\rm\,x\,$ is transcendental over $\rm\,R\ $ (an "indeterminate" in old-fashioned language),$\ $ i.e. $\rm\, x\,$ isn't a root of any polynomial with coefficients in $\rm\,R\,$. One may view $\rm\,R[x]\,$ as the most general ring obtained by adjoining to $\rm\,R\,$ of a universal (or generic) element $\rm\,x,\,$ in the sense that any other adjunction $\rm\,R[s]\,$ is a ring-image of $\rm\,R[x]\,$ under the evaluation homomomorphism $\rm\, x\to s\,.\ $
For example, if $\rm\,R \subset S\,$ are fields then $\rm\,R[s]\cong R[x]/(f(x))\,$ where $\rm\,f(x)\,$ is the minimal polynomial of $\rm\,s\,$ over $\rm\,R\,.\,$ Essentially this serves to faithfully ring-theoretically model $\rm\,s\,$ as a "generic" root $\rm\,x\,$ of the minimal polynomial $\rm\,f(x)\,$ for $\rm\,s\,.\,$
Polynomial rings may be characterized by the existence and uniqueness of such evaluation maps ("universal mapping property"), e.g. see any textbook on Universal Algebra, e.g. Bergman.