dU=dQ-dW
means that the change in internal energy U, is equal to the heat input, dQ (into the system) less the work done by the system, dW. U is a characteric of the system under study, but dQ and dW are not. dQ and dW represent energy transfers.
In a banking analogy, dQ and dW might be deposits and loans by the bank - money in and out - while U would be the amount of money inside the bank. (In this analogy, a withdrawel would be a negative dQ, and a loan paid back would be a negative dW.)
U is changed by dQ and dW, but dQ and dW do not tell us anything directly about the state of the system itself.
dW can be written in terms of state variables. For a PVT system, dW=PdV. If no irreversible processes occur, dQ can be written as TdS, where S is the entropy. Then the first law reads
dU = TdS - PdV.
This is called "the first law for states" to emphasize that it does not work when irreversible (i.e. real) processes take place.
Defining the enthalpy, H=U+PV leads to another form of the first law:
dH = TdS + VdP
Defining the Helmholtz free energy, F=U-TS leads to another form of the first law:
dF = -SdT - PdV
Defining the Gipps free energy, G = U-TS+PV leads to one mor form of the first law:
dG = - SdT + VdP