Thermal radiation is emitted from bodies/gases/objects where the distribution of energy states and particle speeds has reached an equilibrium and these distributions are determined solely by the temperature of the body/gas/object.
In non-thermal radiation, the emission may come from charged particles whose speeds are not determined by temperature (e.g. particles accelerated by magnetic fields - synchrotron radiation, or by electric fields- non-thermal bremsstralung from an X-ray target medium) or where the populations of energy states are not determined by temperature (e.g. a population inversion in a laser).
EDIT: There is some confusion about the first paragraph. What depends solely on the temperature is the distribution of energy states and particle speeds, not the radiation spectrum. The radiation spectrum can also depend on the density, composition and geometry of the thermal emitter. This is largely how we are able to find out information about the universe from radiation - for example the composition of stars, which are to an excellent approximation sources of thermal radiation. Note also that thermal radiation isn't a synonym for black body radiation. For the latter, no information can be obtained from its featureless spectrum other than the source temperature and an emitting area.