What is the meaning of the delay in a FIR filter
In mathematical terms, time discrete systems are are most often expressed in terms of difference equations. A delay is simply $y[n]=x[n-1]$ so the difference equation for your FIR filter would be $$y[n]=\sum_{i=0}^{M}b_{i}\cdot x[n-i]$$.

See https://ccrma.stanford.edu/~jos/filters/ for a good primer