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If you can imagine an atom, of an element, as a sphere, then the radius of that sphere would be the atomic radius of that element. However, as atoms are things far better described using quantum mechanics than classical physics, the definition – even the concept – of atomic radius is tricky (and, in fact, there are actually several different definitions!).
Start with the Bohr atomic model, and an atom of hydrogen. In this model, the atomic radius (Bohr radius) is related to the lowest energy level of the electron, and has an exact value which involves Planck’s constant, the fine structure constant, c (speed of light), and the mass of the electron ( h/(2πcαme) – approx 53 pm … that’s picometers, trillionths of a meter, in case you were wondering). Although the Bohr model of the atom is no longer used, except in teaching, the Bohr atom radius for hydrogen is a key physical constant.
If you have a crystal, of a salt, you can study it with x-rays, and work out how far apart the entities in the crystal lattice are; those entities are ions (not atoms), so the atomic radii estimated this way are called ionic radii. No surprise that the ionic radii of a particular element depend on the ionization state of the ions!
In a metal, one or more outer electrons become part of the sea of electrons throughout the metal, which give the metal its high electrical conductivity. The atomic radii of metal atoms in this environment are called metallic radii.
By now you should be able to guess what the covalent radius is (in molecules with covalent bonds, the atomic radii are estimated from the bond lengths), and what the Van der Waals radius is (if two atoms are not bound in a molecule, the minimum distance between them is determined by the Van der Waals force, and radii estimated this way are …).
Chemguide, a UK site, has a nice intro, Atomic and Ionic Radius; this Frostburg State University page is a bit more advanced ; and here is a list of the elements, sorted by atomic radius.
Astronomy Cast episodes relevant to this topic include Quantum Mechanics, Wave Particle Duality, and Inside the Atom.