What is an irreversible reaction?
The term "irreversible" means that the reverse reaction occurs so rarely that it is considered negligible. This means that you do not have to consider equilibrium, as you have to for reversible reactions. Instead, you can assume that all of the reactants will eventually become product.

As you stated, this is true for reactions that have a very negative Gibbs free energy.  Remember the formulation of the Gibbs free energy in terms of the equilibrium constant:
$$
\Delta G^o =  -RTln(K)
$$

This can be rewritten as follows:
$$
K = e^{-\Delta G^o/RT}
$$

As $\Delta G^o$ gets very negative, you can see that $K$ gets very positive.

The equlibrium constant has two definitions that are both useful. First, it is the ratio of the products to the reactants. Therefore as $K$ gets very positive, the equilibrium ratio of products to reactants approaches infinity, implying that all of the reactants will be consumed.

More useful is the definition of the equilibrium constant as the ratio of the rates of the forward and reverse reaction. From this definition we can see that as $K$ gets very large, the ratio of the rates of the forward and reverse reaction approaches infinity. This means that the rate of the reverse reaction becomes negligible compared to the rate of the forward reaction. This is exactly what it means to be irreversible.

Note that it is possible to make (what I think is) a more convincing argument using the concept of activation energy, using Hammond's Postulate to describe the character of the transition state. If you'd like, I can write this out. However, I think the argument given above is more easily understood with a basic chemistry background.