What is the element of this set? If K is any field. What is the elements of the sets
$$K[x,y]$$ and $$K[x,y]/(x^2,y^2)$$ where $(x^2,y^2)$ is the ideal?
The elements of $K[x,y]$ are polynomials in the variables $x$ and $y$ with coefficients from $K$. For example, $x + xy + y^2$.

The elements of $K[x,y]/(x^2,y^2)$ are cosets of the ideal $I$ generated by $x^2$ and $y^2$. For example, $x + xy + y^2 + I$. In this ring, the elements $x + xy + y^2 + I$, $x + xy + I$, and $x - x^2y + xy + I$ are all equal, since $y^2$ and $-x^2y$ are in $I$.

Now every element $p(x) + I$ in $K[x,y]/(x^2,y^2)$ is equal to a unique element of the form $ax + by + cxy + d + I$, where $a,b,c,d$ are in $K$. It's sometimes useful to identity the elements of $K[x,y]$ with $\{ax+by+cxy+d\mid a,b,c,d\in K\}$, remembering that multiplication is carried out modulo $I$ ("$x^2$ and $y^2$ are $0$"). What we've done here is picked a canonical representative for each equivalence class. This is analogous to how it's sometimes useful to identify the elements of $\mathbb{Z}/n\mathbb{Z}$ with $\{0,1,\dots,n-1\}$, remembering that addition and multiplication are carried out modulo $n$.
$K[x,y]$ are all polynomials in $x$ and $y$ with coefficients in $K$.

$K[x,y]/(x^2,y^2)$ is just a quotient ring. Roughly speaking, a property of that quotient ring is that all terms in the polynomial that have $x^2$ or $y^2$ become zero in the quotient ring.