parameter
A parameter is a knob you set before you start. Within a single problem it stays fixed, so it behaves like a constant; but you imagine turning the knob to get a whole family of related problems. In the line y = mx + b, the letters m and b are parameters: for any chosen m and b you get one particular line, and varying them sweeps out every line there is.
So a parameter sits between a constant and a variable. Relative to the variable x, the parameters m and b are held steady while x runs free. Relative to the family of all lines, m and b are the dials that pick out which member you mean. Whether a letter counts as a parameter is a matter of role: it is the quantity you treat as fixed for now but think of as adjustable across cases.
This dual nature is what makes parameters so useful. Solving ax + b = 0 for x once, in terms of the parameters, gives x = −b/a — a single answer that resolves the whole infinite family of such equations at a stroke, valid for every a (except a = 0, where the equation is not of this form). Reasoning with parameters is how you solve many problems at once rather than one at a time.
In y = mx + b, set the parameters m = 2, b = 1 to fix the line y = 2x + 1; change them to m = −1, b = 4 and you have a different line, y = −x + 4.
Fix the parameters to pick one line out of the infinite family.
Parameter vs. variable in one sentence: the variable is what changes inside one problem; the parameter is what you hold fixed within that problem but vary from one problem to the next.