Why do we use dimensionless expressions in flight mechanics and aerodynamics?
Here is a dimensionful equation: $$L = \frac{\rho}{2}\cdot v^2\cdot \frac{2\cdot\pi\cdot b^2}{1+\sqrt{1+\left(\frac{b^2}{2\cdot S}\right)^2}}\cdot\alpha$$
Note that all ingredients are physical, measurable values. Now here is the same thing again, now in dimensionless form:
$$c_L = \frac{2\cdot\pi\cdot AR}{1+\sqrt{1+\frac{AR^2}{4}}}\cdot\alpha$$
The dimensionless version is independent of speed $v$ and air density $\rho$ and also needs only the aspect ratio $AR$ and the angle of attack $\alpha$ to give a result, while the dimensionful version needs wingspan $b$, wing area $S$ and angle of attack. However, while the dimensionful version yields an actual force, the dimensionless version yields a coefficient which needs to be multiplied with speed, density and wing area again in order to result in a force.

By working in the dimensionless domain, engineers can work on a generalized form of the problem. This was especially helpful in the time before computers: The complex calculation to arrive at a coefficient was needed only once, and computing the dimensionful value from there was trivial.

Also, when comparing between different airplanes, or between wind tunnel models and the real thing, with dimensionless coefficients the numbers will be very similar, which allows much better to error-check results or to make first assumptions on a new design.

However, there are two drawbacks:


You need to know the reference values! It is easy to screw up, e.g. when combining values from the wing and the tail which are referenced to their individual area. While lift can be added right away, the lift coefficients must first be converted to a common reference value.
Sometimes thinking in coefficients can be misleading: Just consider induced drag. If written in dimensionless coefficients like this
$$c_{Di} = \frac{c_L^2}{\pi\cdot AR}$$
you can be forgiven for thinking that induced drag goes down with increasing aspect ratio. But this is wrong! Now the same again with dimensions:
$$D_i = \frac{L^2}{\frac{\rho}{2}\cdot v^2\cdot\pi\cdot b^2}$$
This shows that induced drag at the same speed is inversely related to span loading, the amount of lift created per unit of wing span.