In the United States generally and to a much lesser extent in the UK and a few other countries you’ll see some very old-school units of measurements. Miles, yards, pounds, fortnights, pints, gallons, and numerous others. Most of the rest of the world uses metric units, the primary variant of which is called SI units used almost universally in the physical sciences. You know, meters, kilograms, that sort of thing. My American patriotism aside, SI is of course a lot easier to work with. Kilometers to meters is a lot easier than fathoms to nautical miles (or whatever).
But it’s not perfect. In a sense it would be more natural to define units of measurement in terms of physical constants rather than physical constants in terms of whatever human-sized conveniences happen to be in use. For instance in electrostatics we have (as an example) the force between two charges:
Where the q are the charges in the usual SI unit of coulombs, and that epsilon is the electric constant (itself often called the vacuum permittivity). The permittivity is some emperically measure decimal value with no clean form.
Or you can pick out a different system of units. In Gaussian units the basic unit of charge is finagled in such a way so as to cancel out that ugly proportionality constant, like this:
This is a choice much beloved of theorists, for good reason. Nothing of theoretical interest is gained by carrying around awkward proportionality constants, and sometimes they even manage to obscure the fundamentals of the theory. My E&M instructor this semester is using Gaussian units, and in the classical field theory course he teaches he goes farther and sets c = 1. Newer editions of Jackson apparently use standard SI, much to his dismay.
In other branches of physics this sometimes gets taken to extremes. You’ll see things like G = 1, h = 1, e = 1 (the electron charge, not Euler’s number), α = 1 (incompatible with c = 1, however), and a host of other possibilities.
I should note that all this is still technically metric, but it’s not SI. In classes like this one we so rarely use actual numbers that it makes little difference. For the purposes of actually getting numbers in order to compare to experiment you simply do the conversions as the very last step. No point in dragging in weird constants before that. The classical field theory class I mentioned was actually something I took last semester – I used a calculator exactly once.