...General relativity and pure quantum mechanics have no dimensionless constants, because the speed of light, the gravitational constant, and Planck's constant merely suffice to set units of mass, length and time. Thus, all the dimensionless constants come in from our wonderful, baroque theory of all the forces other than gravity: the Standard Model.
For starters, we have a bunch of masses. There are 6 kinds of quarks, one positively charged and one negatively charged of each generation: up, down; charmed, strange; top, and bottom. The masses of these quarks, divided by the Planck mass, give 6 dimensionless constants. We also have 3 kinds of massive leptons --- electron, muon, tau. The W and Z bosons also have their masses. Then there is the Higgs, which while still not detected, is very much part of the theory, so we get another mass.
This gives us 6 + 3 + 2 + 1 = 12 dimensionless constants so far.
Then we have two coupling constants: the electromagnetic coupling constant and the strong coupling constant. The electromagnetic coupling constant is just another name for the fine structure constant; it describes the strength of the electromagnetic field. Similarly, the strong coupling constant describes the strength of the strong force - the force transmitted by gluons, which binds quarks together into baryons and mesons.
You may wonder why I'm not listing a coupling constant for the weak force here. The reason is that you can calculate this from the numbers I've already listed.
I should warn you here: there are different ways of slicing the pie. Instead of the electromagnetic coupling constant together with the masses of the W, Z, and Higgs, we could have used 4 other constants: the U(1) coupling constant, the SU(2) coupling constant, the mass of the Higgs, and the expectation value of the Higgs field. These are the numbers that actually show up in the fundamental equations of the Standard Model. The idea is that the photon, the W and the Z are described by an U(1) x SU(2) gauge theory, which involves two coupling constants. The beautiful symmetry of this theory is hidden by the way it interacts with the Higgs particle. The details of this involve two further constants - the Higgs mass and the expectation value of the Higgs field - for a total of 4. If we know these 4 numbers we can calculate the numbers that are easier to measure in experiments: the masses of the W and Z, the electromagnetic coupling constant, and the mass of the Higgs. In practice, we go back backwards and use the constants that are easy to measure to determine the theoretically more basic ones. ...