Looking at how linear feedback shift registers (LFSRs) are
implemented, one can notice that the tap coefficients feeding into the
XOR have to be from an irreducible polynomial in the polynomial ring
$\GF(2)[x]$. Why is this?
Suppose we have polynomials modulo $x^2+x+1$. These polynomials form
the Galois field $\GF(2^2)$. An LFSR shifts, which means it takes a
polynomial $p(x)=p_0+p_1x$ representing the current $2$--bit state
vector, and multiplies it by $x$. After shifting, the highest order
term $p_1x^2$ needs to be folded back into the representation by
eliminating $x^2$ using the equation $x^2 + x + 1= 0$, which follows
from modulo arithmetic. Hence, LFSR is shift followed by conditional
XOR, which means addition of $x+1$ if $p_1 = 1$.
So, the shift represents multiplication by the generator polynomial
$x$ which produces a sequence of all elements of the field's cyclic
multiplicative group. The XOR computes the modulo operation after
multiplication, and is there to keep the representation contained in
the minimal amount of bits.
%[1] http://en.wikipedia.org/wiki/LFSR