van der Waals equation
The ideal gas law pretends molecules are sizeless points that never attract one another. The van der Waals equation is the famous first patch on that fiction: it takes PV = nRT and adds two small corrections so that the equation describes real gases more faithfully, especially when they are squeezed or chilled.
It makes two physical fixes. First, because molecules do take up room, the space they have to roam in is a little less than the container's volume, so a term is subtracted from the volume. Second, because molecules gently attract one another, the pressure on the walls is a touch lower than the simple theory predicts, so a term is added back to the pressure. Each gas has its own pair of constants, a and b, measuring the strength of attraction and the size of its molecules.
The equation matters as the clearest bridge from the ideal-gas idealisation to real-gas reality. It is not perfectly accurate, but it captures the essential physics — including, strikingly, the existence of a critical point and the way a gas can condense into a liquid, things the ideal gas law can never describe.
Compress carbon dioxide hard at room temperature and the van der Waals equation correctly predicts that it can turn into a liquid — a phase change the ideal gas law, with no attractions built in, is utterly blind to.
Built-in attractions let the equation predict a gas condensing to liquid.
The van der Waals equation was an early and clever improvement, not the last word. More accurate equations of state exist; van der Waals is prized chiefly because its two correction terms have such clear, intuitive physical meanings.