See my writeup on stimulated emission.
Spontaneous Emission is the process by which an excited electron decays to a lower energy state in the absence of external stimulation.
Electron energy states have a lifetime associated with them that indicates how long they will stay on average in an excited state. In general the number of electrons in an excited state will decay according to the law: N = Noe-t/tau where tau is the characteristic lifetime. There are a number of variables that will affect this time and a quantum mechanical analysis is necessary to understand them in general. I will not get into the quantum stuff because it's hard to node and I don't feel that I understand it well anyway.
There are two different ways in which decay or relaxation can occur: radiative and nonradiative. In radiative, energy is released in the form of photons that have an energy equal to the difference between the upper and lower energy levels that the electron moves between. The color is of course also determined by this energy difference. The phase of the photon in spontaneous emission is random as is the direction the photon propagates in. This is not true for stimulated emission.
In nonradiative relaxation, the energy is absorbed as phonons, more commonly known as heat. Nonradiative relaxation is nearly impossible to measure and can only be inferred except in very small particles because the difference in the temperature before and after a relaxation is so small that it is in the noise.
Nonradiative relaxations are typically for smaller transitions than radiative relaxations. For many materials (for instance semiconductors), electrons move quickly from a high energy level to a meta-stable level via nonradiative transitions and then move down the the bottom state via a optical or radiative transition more slowly. This is because crystal lattices generally can not support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of lasers. Namely since electrons decay slowly from them, they can be piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.