I guess, you are referring to the mix columns operation from the point of view with polynomials over $GF(2^8)$.
A detailed explanation of this can be found at Wikipedia. It is a quite unusual structure, but it behaves just like you would consider any $GF$.
- The columns are first considered as polynomial, or better: as coefficients for a polynomial of degree 3.
- Then this polynomial is multiplied with a constant polynomial ($3x^3+x^2+x+2$)
- Addition in this field is done with the XOR operation
- Multiplication is quite complex, for details check out this.
As for your questions:
There are two different modulo operations. One is considered on the $GF(2^8)$, which is $100011011$. But this is just the finite field for the coefficients, and in the overall structure you actually deal with polynomials modulo $x^4+1$. I guess you have an error in your actual number there, because if you just write the hexademcimal representation of the coefficients, that would be the coefficients $1,0,0,1$ (with coefficients from 0 to 255), or $1|00|00|01$ written in hexadecimal.