An understanding of what is a coupling constant is essential aspect of the notion
of unification. A good example is electricity and magnetism. It was at first
thought that a universal constant
is required in the force
laws between charges, and a separate universal constant 
for
the force law between magnets. But later it was realised that magnetism and
electricity are related phenomena depending on motions of charges and magnets.
It was found that the product
in free space
gives c2, the square of the speed of light, a universal constant.
Thus there was only one coupling constant needed to describe electricity and
magnetism, and the latter two were "unified".
Minimizing the number of phenomenological parameters is at the heart of the predictive power of theories. To describe properties of real substances we need a large number of phenomenological parameters. But most of these can be ascribed to the particulars of the state in which we find the substance. For instance the elastic moduli of a crystal can be ascribed to the lattice arrangement of the ions, th shape of the electron clouds in individual ion which determines its interaction with its neighbors and so on. But the fundamental interaction involved is Electromagnetism, determined by one coupling consonant. As is well known, at the present state of knowledge there are four interactions we may call fundamental. The Strong Nuclear, Weak Nuclear, Electromagnetism and Gravitation. The hypothesis of unification is that the need for four different coupling constants is a consequence of the particular "state" in which we find ourselves. If only we could see the specific conditions characterizing this state, we could progress towards identifying the universal principle.
Another prime example is Newton's Law of Universal Gravitation. With the benefit of hindsight this may appear trivial even to school students. But no one suspects at first that the fall of an apple on the earth is governed by the same law that governs the orbital motion of the moon around the earth, and indeed the motion of all "heavenly" bodies. One may introduce one constant g as the constant acceleration due to gravity on the surface of the earth, and another constant G to account for the lunar motion. But g is a derived constant including the specifics of earth radius and mass, and only one universal constant G is needed to give the Gravitational force law.