I know that in an adiabatic expansion, $W = -U.$ My question is, is the work positive or negative? I'm confused on the difference between work done "by" the system and work done "on" the system.
The most common expression for the first law is
$$\Delta U=Q-W$$ where
$W$ is positive if work is done by the system (work out, energy out) and negative if done on the system (work in, energy in). An expansion is considered work done by the system.
But sometimes in chemical thermodynamics it is expressed as
$$\Delta U=Q+W$$ where
$W$ is negative if work is done by the system (work out, energy out) and positive if done on the system (work in, energy in).
It doesn't matter as long as you are consistent. Think of like this. If energy is going out of the system, there will be a reduction (negative change) in internal energy, and vice versa. Both versions are consistent with this. And of course, $Q=0$ in both cases for an adiabatic process.
Hope this helps
Work done $by$ the system is positive. This is just a convention and interestingly the opposite convention about the sign of work is taken in chemistry.
Work is equal to the product of force and displacement. So if the force that A exerts on B is in the same direction as the displacement of B, the work that A does on B is positive. If the force that A exerts on B is opposite to the direction of B's displacement, then the work that A does on B is negative.
In the equation that you have written, W is the work that the system does on its surroundings. It can be either positive or negative in sign, depending on whether the gas expands or is compressed, respectively.