What do the filter coefficients in a digital filter represent?
We can try a very short introduction:


Every filter represents a Linear Time Invariant System (LTI)
Every Linear Time Invariant System can be completely described by it's transfer function or it's impulse response. The two can be converted into each other by the Fourier Transform
Filter coefficients are derived from impulse response or transfer function
The exact nature of the filter coefficients depends on the algorithm (there are quite a few of those)
In the case of the simplest algorithm, the direct convolution FIR (Finite Impulse Response) filter, the filter coefficients are simply the impulse response of the LTI system.
In most other algorithms the relationship is much more complicated and text book study is indeed required.
The whole subject of LTI systems, transfer functions, Fourier Transforms, amplitude responses, phase responses etc. is probably another text book worth of stuff