Importance and Intuition of Polynomial Rings I know that a polynomial ring $R[x]$ is the ring with elements consisting of polynomials with coefficients in $R$. However, this definition leaves me confused when I try to really understand the concept of polynomial rings, rather than just accept that definition for what it is. Where did the polynomials come from? Why are polynomial rings important? Is there a way to “construct” the polynomials in the context of Abstract Algebra, so that we may see why studying rings made up of polynomials is important? Are polynomial rings actually just sets of polynomials, or are the polynomials just a concrete way to represent some more abstract idea? What does the variable $x$ mean in polynomial rings? I apologize for all the questions, I’m just having trouble seeing the importance or intuition behind them.

$R[x]$ can be thought of as the simplest way to add one element to the ring $R$ and still have a ring.
Suppose I have a ring $R$ and I want to add one extra element which is not already in $R$. Call that extra element $x$. Consider the set $A = R\cup \{x\}$. To make $A$ into a ring, we need to add in all sums of terms that look like $r x^k$ where $r\in R$ and $k\in \mathbb{N}$. The simplest way to define multiplication in this ring is just to let multiplication of two members of $R$ stay the same and let all the products $r x^k$ be unique elements of the new ring. This is just the polynomial ring $R[x]$.
(This is actually just a restatement of the comment by @lhf, but it is likely that OP does not know what a universal $R$-algebra is.)
The philosophy behind polynomial rings is that we are not committing ourselves to saying exactly what $x$ represents. It is just an "indeterminate" and we can manipulate it according to formal rules like $x^{a} x^{b} = x^{a+b}$. If at some point we wish to specify a value for $x$, we can do so using a ring homomorphism $R[x] \rightarrow R$ that replaces every instance of $x$ with that particular value.
Note also we can define an homomorphism from $R[x]$ to the ring of all functions from $R$ to $R$ with pointwise operations. Just send $r \in R$ to the constant function $r$ and send $x$ to the identity map on $R$. In the case $R = \mathbb{R}$, this map is an isomorphism onto its image. This is how we get the correspondence between polynomial rings and polynomials as we know them from grade school algebra.
