Misha's comment could be a bit misleading. In particular, it is not true that the Ricci flow should exist on a slightly bigger interval $(-\epsilon,T)$ with $g(0) = g_0$. One way to see this is by thinking about a theorem of Bando (see http://www.springerlink.com/content/v0764574t4764138/ if you have access) which says that if $(M,g_t)$ is a solution for Ricci Flow on the interval $[0,T)$, then $g_t$ is real analytic with respect to the normal coordinate charts on $M$ for $t>0$. In particular, if $g_0$ was not real analytic, we cannot extend the flow backwards for any $\epsilon>0$ (the comment was only in charts, but by compacntess if we can do it in charts, we can do it on the whole manifold for some small $\epsilon>0$) because then Bando's theorem would imply that $g$ were real analytic.

The correct statement is just like for the heat equation. We say that $f$ is a solution to the heat equation
$$
\frac{\partial f}{\partial t} = \Delta f
$$
on $[0,T)$ if the above equation is satisfied for $t>0$ and $\lim_{t\searrow 0} f= f_0$. In particular, there is no "meaning" of the equation at $t=0$, only for $t>0$. Do not get confused by trying to apply ODE intuition to the PDE. Parabolic equations are not like ODE's in the sense that you can just "go in the direction of $\Delta f$".

So, for completeness, here is what it means to be a solution to RF on the interval $[0,T)$ with initial data $g_0$:

The metric $g_t$ is smooth for $t\in (0,T)$ and for such $t$, $g_t$ satisfies
$$
\frac{\partial g_t}{\partial t} = -2Ric_{g_t}.
$$
You can think of this either in local coordinate charts, as Misha does, or just as a coordinate free equation for the symmetric 2-tensor $\frac{\partial}{\partial t} g_t$.

Furthermore, we require that $g$ is continuous up to $t=0$ (because here we're only interested in solving RF with smooth initial data, if we wanted to start with rough data, we'd require a limit in some sense) and
$$
g_{t=0} = g_0
$$
at each point in $M$.

If you're confused, you should read up on the heat equation first. Its *exactly* the same. In particular, after reading about the heat equation, you should read about the De Turk trick, which transforms the RF into a strongly parabolic equation (i.e. heat-type equation) by fixing the diffeomorphism gauge. A quick google suggests the following chapter http://www.springerlink.com/content/0t673151r72133r7/ as a possible reference. Any book on Ricci Flow should have a good description of this.