What is Galois Field

Galois field is the name that engineers (and especially those studying error correcting codes) use for what mathematicians call a finite field. In applications, the most commonly used Galois field is $\text{GF}(256)$, also called $\text{GF}(2^8)$. Its elements can be thought of as polynomials of degree $7$ or less with binary coefficients ($0$ or $1$). Addition of two field elements is addition of the two polynomials with coefficients being added modulo $2$. Multiplication is polynomial multiplication modulo a polynomial $m(x)$ of degree $8$, that is, multiply the two given polynomials (which may result in a polynomial of degree as much as $14$) and then divide by $m(x)$, throwing away the quotient and keeping only the remainder.

A Galois field is a finite field (from the Wikipedia article):

In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements.