I decided to learn analysis over the summer for fun, but I'm really confused by the field properties. Why is the 2 element set of 0 and 1 a field? Addition wouldn't be satisfied, because 1 + 1 = 2, which isn't in the set.
Also, if there are a set of properties that completely determine the real number system, why does that imply that there is only one? My book says something about a one to one correspondence between the reals and some other real system that preserves the functions of + and *, but I don't understand this at all.
The field $\mathbb{F}_2$ is indeed a field. The important thing is to remember that addition is done modulo $2$. Thus,
$$ 1 + 1 = 2 \equiv 0 \pmod{2}, $$
and $0$ is indeed in $\mathbb{F}_2$.
Furthermore, the real numbers are the unique complete totally ordered field, and they can be constructed by say, completing the rational numbers with the metric $d(x,y) = |x - y|$, where $|\cdot|$ is absolute value. If you share with us what is written in the book you are using, we may be able to help a bit more.