It's also a "fudge factor" so that our equations work out. For example, we all know E = mc2, but many physicists set c = 1 so that E = m and the fudge factor disappears. I've often wondered if it was possible to develop a mathematical/physical system where all of the fudge factors for the fundamental constants of the universe could be eliminated, but my mathematical expertise is nowhere near that sophisticated.
It depends on the context. The word "constant" have connected but different meanings in maths and in physics. As you're posting in the physics forum, I guess you're more interested with the physics notion. In a physical theory a constant is a quantity used to describe the behaviour of a system (together with plenty of other quantities) but that does not depend on which system is studied, what are its environement, the interval of time considered or the position in space of the system. So its value can be given (whithin a given system of units) independently of any experience.
In this case "k is a constant" means that k is a number, whose value does not depend on the mass, acceleration, force or any other physical properties of the system. I.e. whatever you apply it to, "k" will have the same value always (if you found it to be 2 by some experiment, for example, you could write F = 2ma).
Note that another "kind" of constant is for example where we take F = ma, and I say "a car is moving along a road with constant acceleration". In that case, I mean that a has some specific value which will not change in this specific situation that I am considering - obviously I don't mean that the car can never have another acceleration, nor that all the other objects in the universe must have the same value of a.