An expression with multiple terms, each a different power of the same variable. The standard form is:
$$c_nx^n + … + c_1x + c_0$$
The variable is $x$. The $c$'s are coefficients of $x$; $c_0$ is the constant term. The degree $n$ is the value of the exponent of the highest power term. $A$ mod $2$ polynomial of degree $n$ has $n+1$ bits representing the coefficients for each power: $n, n-1, ..., 1, 0$.
Perhaps the most insightful part of this is that the addition of coefficients for a particular power does not "carry" into other coefficients or columns.