It is 'spontaneous' because we cannot 'make it happen' by changing the conditions the sample is in - such as making it hotter or putting it under pressure. Similarly we cannot stop it happening - that is why nuclear waste is such a problem!
The emission of the nuclear radiation is a purely random event. It cannot be predicted exactly when an atom will decay, only that a certain number will decay in a given time. The mathematics of probability is used for this requires a large number of atoms to be considered. (See half life and radioactive decay series).
Radioactive nuclear decay occurs whenever a nucleus is in an energy-state that is not the lowest possible for its nucleon number. This state may occur naturally (which essentially means that it was created in that state when formed within a star) or by artificial means (neutron or photon irradiation).
The rate of decay depends on the number of undecayed nuclei present, so with each decay event there is a decrease in the activity of a radioactive sample.
See here for the dangers of nuclear radiation.
For a more mathematical look at this - see here