Simply put (more or less already said above). Let $K=C(x,y)$ and let $xi(t)$ be a generalized power series (a power series with well-ordered exponents) where the exponents are non-negative real numbers. Assume (this is important) that $P(t,\xi(t))\neq 0$ for any $P\in K$. Then the map: $\nu(P(x,y))=ord_{t}(t,\xi(t))$ is a valuation (the "order of contact of P with $(t,\xi(t))$").

Notice that if $\xi(t)=t^{\pi}$, for example, the valuation has rational rank $2$. If $\xi(t)$ corresponds to an analytic branch of a curve (non-algebraic), the valuation has rational rank $1$, etc.

By the way, here you may find something useful. That book may be of help.

You should read something about point blowing-ups and then you understand the projective limit thing. But without it, it gets somewhat too algebraic.