The process by which an electromagnetic field causes an electron in an excited state to fall to a lower state and emit a photon in phase with the original field. Lasers are designed to take advantage of this process.
An atom's properties are mostly described by its electronic configuration. Stimulated emission is really a quantum mechanical phenomenon but it can be understood in terms of a classical field and a quantum mechanical atom.
A common joke in my undergraduate physics classes was that if you didn't know what to put on an essay problem or when asked something in class, "The interaction of electrons" was almost always a vague but correct answer. ( I'm a geek and I like to hang out with geeks.) Electrons and how they interact with each other and electromagnetic fields form the basis for most of our understanding of chemistry and physics. Electrons have energy in proportion to how far they are on average from the nucleus of an atom. The Pauli Exclusion Principle forces some electrons to be further from the nucleus than others (that's why all the electrons don't just hang around in the 1s orbital.) When electrons absorb energy either from light (photons) or from heat (phonons), they move further away from the atomic nuclei but they are only allowed to absorb energy that will land them into specific energy levels. This leads to emission lines and absorbtion lines.
When an electron is excited, it will not stay that way forever. On average there is a lifetime for any particular energy level after which half of the electrons initially in that state will have decayed into a lower state. When such a decay occurs, the energy difference between the level the electron was at and the new level must be released either as a photon or a phonon. When an electron decays due to "timeout" it is said to be due to "spontaneous emission." The phase associated with the photon that is emitted is random and has to do with some quantum mechanical ideas concerning the atom's internal state. If a bunch of electrons were put into an excited state somehow and then left to relax, the resulting radiation would be very spectrally limited (only one wavelength of light would be present) but the individual photons would not be in phase with one another. Other names for this process include fluorescence, relaxation transitions, and energy decay.
Other photons (i.e. an external electromagnetic field) will affect an atom's state. The quantum mechanical variables mentioned above are changed. Specifically the atom will act like a small electric dipole which will oscillate with the external field. One of the consequences of this oscillation is it encourages electrons to decay to the lower energy state. When it does this due to the presence of other photons, the released photon is in phase with the other photons and in the same direction as the other photons. This is known as stimulated emission.
Although most directly related to the discussion of how lasers work, stimulated emission touches on some of the most basic concepts in physics and the interaction of light and matter. It is a very important and key understanding to the understanding of optics specifically and physics in general.