What a gas is holding: internal energy
Zoom into a gas and you see molecules flying, spinning and (in molecules) vibrating, ceaselessly. Add up the kinetic energy of all that microscopic motion, plus any potential energy of the forces between molecules, and you get the internal energy U of the gas. It is a genuine store of energy that lives inside the system — unlike heat and work, which only ever describe energy crossing the boundary.
For an ideal gas there is a beautiful simplification: the molecules are so far apart that their mutual potential energy is negligible, so U depends only on temperature — not on pressure or volume. Temperature is the internal energy of an ideal gas in disguise. How much energy per molecule depends on the degrees of freedom: a single atom can only move in three directions, giving three ways to store energy.
For a monatomic ideal gas (helium, argon), internal energy is exactly three-halves of nRT — purely the translational kinetic energy of the atoms.
More generally, a change in internal energy is proportional to the change in temperature, with the constant CV (the molar heat capacity at constant volume) counting how many ways the molecule can store energy.
Two doors into the system: heat and work
There are exactly two ways to change a system's internal energy, and both act at the boundary. Heat Q is energy that crosses because of a temperature difference — the disorganized jostling of hotter molecules nudging cooler ones. Work W is energy that crosses through an organized force acting through a distance — most often a gas pushing a piston out, or the surroundings pushing it in.
The first law: energy bookkeeping
Put the two doors together and you get the first law of thermodynamics — nothing more than conservation of energy, now that heat is counted as energy. Whatever heat you pour in either stays as internal energy or leaves again as work. Nothing vanishes.
The first law: the change in internal energy equals the heat added TO the system minus the work done BY the system. Q > 0 means heat in; W > 0 means the gas pushes outward and does work on its surroundings.
Why U is special: state versus path
Here is the subtle point that separates the first law from everyday intuition. Internal energy U is a state function: it depends only on where the gas is now (its P, V, T), not on how it got there. Heat Q and work W are the opposite — they are path-dependent. Take a gas from state A to state B by two different routes on the PV diagram and you will generally supply different Q and get different W — yet \Delta U, their difference, comes out exactly the same both times.
A first calculation
Let us use the law right away. Suppose you add 200 J of heat to a gas in a cylinder, and during that heating the gas expands and pushes the piston, doing 80 J of work on the surroundings. How much did its internal energy change?
Of the 200 J poured in, 80 J immediately left again as work on the piston; only the remaining 120 J stayed behind, raising the gas's internal energy (and hence its temperature).
Notice how the whole thing is just careful accounting: in minus out equals change stored. Every problem in the rest of this track is a variation on that one sentence — the hard part is only figuring out Q and W for the particular process, which is where PV diagrams come in next.