It does whatever the patch is designed to interpret it as. For '80s style synths, it's usually controlling the frequency or amplitude of the LFO, or perhaps the distortion amount. For an organ patch it might control the rotary speed. For a string patch it might control the frequency or amplitude of vibrato.
Modulation is just one of many MIDI control types, so patches might have a variety of things mapped to different MIDI controls. Volume is its own control, and commonly keyboards will have a slider dedicated to it.
Here's a full list of the MIDI control types that are defined in the MIDI spec.
In an old-school setup you'd want a keyboard with as many wheels, sliders, and knobs as possible so you could set up a whole bunch of stuff. In modern setups you're usually doing remapping in software, so you can have one physical control do different things depending on where you are in your performance, so you don't need as many physical controls.