What's a "delay line"?
The term delay line comes from (analog) telecommunications where timing between different signals needed to be adjusted by delaying one of them. For example, the chrominance and luminance channels in a cathode-ray-tube colour television.

The reason it's called a line is because that's all it really is: a wire. The website linked to above has the following diagram of one.



In the discrete (digital) domain, I've more usually heard delay lines called tapped delay lines.  That's because each sample is a delay, and to form an FIR filter, we need access (to tap into) the signal at different delays.  The line of $z^{-1}$ boxes in the diagram below is what I would call a tapped delay line.