A laser works by exciting the electrons in specific elements or compounds, and amplifying the radiation they give off when they relax.
The most common example of this is the "ruby laser." A rod of ruby crystals is enclosed in a tube with a 100% mirror on one end, and a 95% mirror on the other. A coil of wire is wrapped around the tube and electricity runs through the wire in pulses. During each pulse, the energy excites electrons in the ruby. The electrons briefly jump into higher orbits of the atomic nuclei than normal. Between the pulses, the electrons fall back into their normal orbits, and release their energy as "red" photons. The photons begin bouncing back and forth between the two mirrors until enough build up to overcome the 95% mirror, and these photons are emitted as a laser beam.
The difference between stimulated emission and spontaneous emission, is whether an outside source caused it. Spontaneous emission doesn't happen very often, because the electrons don't naturally jump to higher orbits, they naturally tend to stay in the lowest orbit they can. Stimulated emission is an artificial way of getting the electrons into higher orbits and changing energy from one form (usually electricity) into another (photon emissions).