In this context irreversibility means that once the drug binds with its target, it's done. It can't unbind.

Reversible means that there is an equilibrium of

E + D <-> ED

where E is enzyme and D is drug, and ED is the "complex" molecule. Irreversible binding looks like:

E + D --> ED

where the product on the right is effectively unable to become the molecules on the left again. An example of irreversible binding is COX-2 and aspirin.