path function
Think again of climbing a mountain. Your final altitude does not care which trail you took, but the distance you actually walked — and how tired you got — depends entirely on the route. A path function is a quantity of that second kind: its value depends not just on where you start and end, but on the particular path travelled in between.
In thermodynamics the two great path functions are heat and work. The same change in a gas — say, the same rise in temperature — can be reached by adding a lot of heat and doing little work, or by doing a lot of work and adding little heat. The energy change is fixed because energy is a state function, but how much arrives as heat versus work is not. That is why we write small changes in heat and work without the Δ used for state functions.
Recognising path functions keeps you from a common blunder: you cannot speak of the heat a system contains, only of the heat that flows during a particular process. A system has a definite energy, but it does not have a stockpile of heat or a stockpile of work; those are names for energy in transit along a chosen route.
To warm a room from cold to cosy you might run the heater hard for a short while or gently for a long while; the rooms end up the same, but the heat and electrical work spent along each route differ.
Heat and work depend on the route taken, not merely on the start and end.
Heat and work are path functions, but their sum — the change in internal energy — is a state function. The path dependence of the two parts cancels out, which is the content of the first law.