If for a chemical reaction the standard gibbs free energy is negative then :
All reactions are in some sense equilibrium reactions. The Gibbs free energy is just a convenient measure of energy that we can use to understand more about our system. In particular you may be familiar with a few equations for the Gibbs energy:
\begin{equation}
\Delta G =\Delta H-T\Delta S,
\end{equation}
\begin{equation}
\Delta G=-RT\ln k
\end{equation}
\begin{equation}
\Delta G =-nFE
\end{equation}
For chemists, these three equations are the most commonly used and depend on what exactly you are investigating, but there are plenty of others too.

If we consider the second of these equations we note that $k$, the equilibrium constant, appears. We can see that as the equilibrium constant increases (and since $x$ follows $\ln x$ in limits) we note that the change in Gibbs energy becomes more negative the larger the equilibrium constant becomes. In other words the more products you have relative to reactants, the more negative the change in the Gibbs energy is, we call this spontaneous. 

We could also take the third equation and note that $E$ is the reduction potential for a redox couple ($X^{n+}+e^{-}\rightarrow X^{(n-1)+}$). The more positive the potential for the reduction, the more spontaneous the reaction is.

Hope that helps you, oh and I'm assuming you mean reversible in the reaction sense of equilibria not in the thermodynamic sense of a reversible process?