GF(2) stands for "Galois Field with 2 Elements". It is an example of a finite field:
http://en.wikipedia.org/wiki/Finite_field
You are probably used to using the real numbers as your scalars when you work with matrices (so entries are real and scalar multiplication is by real numbers). It turns out that most or all of the matrix theory you know is valid when you allow scalars from certain other algebraic structures, called "fields". A field is a set with two operations, addition and multiplication, which have all of the nice properties of addition and multiplication that the real numbers have. See
http://en.wikipedia.org/wiki/Field_(mathematics)
for more specifics. In particular, since you can add and multiply elements in a field, then matrices whose entries are drawn from the field can be added and multiplied using the same algorithms you already know.
So...a field is a set with two operations satisfying a bunch of nice properties. Which set is GF(2)? It has 2 elements: {0, 1}. The rules for addition and multiplication are exactly the same as if 0 and 1 were the usual real numbers 0 and 1 (so that 0 times anything is 0, for example), with one exception: since there is no "2" in the set, we define "1+1=0".
To find the rank of a matrix, you can calculate the dimension of the row space with Gaussian elimination. Just remember that the entries are all 0's and 1's with the operations I just described, so for instance to eliminate some 1's below a leading 1, you can simply add rows together since the leading 1's will add to 0 with the operations in GF(2).
One word of caution: it is possible for vectors of 0's and 1's to be independent when considered as vectors with real number entries, but dependent when considered to have entries in GF(2). But if you reduce your matrix to row-echelon form to determine the dimension of the row space, this phenomenon will not arise in that particular case.