Nature itself doesn't have fundamental or derived quantities. We have choices of units with more or less direct ties to things we measure.
In modern systems of units, we try to simplify things by making units for different quantities have simple relations. For example, in both the SI and CGS systems the units for energy (Joule and erg, respectively) are the product of the mass unit (kilogram or gram) times the square of the length unit (meter or centimeter) divided by the square of the time unit (seconds, in each system). That's much more convenient than having some unrelated unit of energy (say a BTU) with some sort of conversion factor to remember.
In principle we could express all of our quantities in units derived from some three starting units- say mass, length, and time. In practice most unit systems in use don't follow through completely on the systematization. For example, temperature, which is basically just an energy scale, is typically measured in Kelvin, which then has to be converted to energy by multiplying by Boltzmann's conversion factor.
Within the systematic part of the units, you can always pick some subset of the units (like the mass, length, and time I mentioned) and derive the other units from them. However, you could also pick different subsets, say energy, length, and speed, and derive the other units from combinations of them. In this example, the mass unit would be the energy unit divided by the square of the speed unit.
So the distinction between fundamental and derived is somewhat arbitrary. There is a distinction between units that are based directly on some measurable quantity (say time as some factor times the period of some atomic emission) and ones that are put together from those measured quantities. Once we've chosen which ones to tie to measurements and which to make from combinations of those, we call the measurement-based ones fundamental. However, those choices change occasionally as measurement techniques change, since we try to keep the units based on the quantities that can be most precisely measured. When that happens, which ones are "fundamental" or "derived" changes.
In very fundamental physics, it's conventional to use Planck units. In these, the speed of light is chosen as the speed unit, Planck's h-bar is chosen as the angular momentum unit, and the third unit needed to form all the others is chosen to be Newton's universal gravitational constant, G. All the other units are then derived from these three fundamental constants.
(published on 06/04/2012)