Particle physicists usually don't use that term much because it doesn't really convey the right ideas -- we have some better terms, and I'll try to explain them.
Everyday matter is made up of atoms, and atoms themselves are made up of protons, neutrons, and electrons. Since the protons, neutrons, and electrons are pieces that make up atoms, they are called "subatomic particles" (somehow being "smaller" than an atom). This fuzzy idea already runs into problems -- a very low-energy electron all by itself in space occupies a wavelike state that can be much bigger than an atom. But somehow the name stuck.
I'm not sure whether photons are subatomic particles or not. Radio waves are made up of photons which can have wavelengths of many meters. Electromagnetic forces, like the ones holding electrons inside of atoms, are conveyed by photons, so you can say that atoms have photons inside them too.
There are lots more particles which really aren't to be found inside of ordinary atoms. Cosmic rays are largely made up of muons, a heavier cousin of the electron. And neutrinos travel at almost the speed of light and don't stay put. You won't find muons inside of atoms -- are they "subatomic"? I'm not sure. There are quarks and gluons inside the protons and neutrons, and heavier cousins of them which can only be produced in very high-energy collisions, where the energies are so high that atoms would not stay bound together.
As far as we can tell, the natural radius of an electron is zero, as are the radii of the quarks and leptons (electron, muon, tau). This is really just a statement that we do not know yet of any substructure inside of these pieces -- perhaps there is some substructure (string loops, or who knows?). We prefer to call these things "elementary particles" since we don't know if they have pieces inside. Lots of exotic stuff can be made up of elementary particles that is also not to be found in atoms, such as a Lambda particle. A Lambda particle is just like a neutron, with one of its quarks (a down quark) replaced by a strange quark. I guess a Lambda particle has a lot in common with a subatomic particle but isn't found in atoms, and it isn't elementary either (it's made up of quarks and gluons). We just call it a particle and don't worry too much.
(published on 08/03/06)