Atomic Radius:
The radius of an atom can only be found by measuring the distance between the nuclei of two touching atoms, and then halving that distance.
Ionic Radius:
It is the radius of an atom's ion. Although neither atoms nor ions have sharp boundaries, they are sometimes treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. Ionic radii are typically given in units of either picometers (pm) or Angstroms (Å), with 1 Å = 100 pm. Typical values range from 30 pm (0.3 Å) to over 200 pm (2 Å).
For further reading: Atomic and Ionic Radius