In short, laser cooling is the process of using a laser to super-cool gases. Duh. Before I get into a deeper explanantion, some definitions to clarify:
Laser - Laser is an acronym for "Light Amplification through Stimulated Emission of Radiation," where the radiation is in the form of photons, the discrete "packets" (quanta) of energy that are the embodiment of the particle nature of light. The light is amplified because a laser is coherent light. That means every photon's wave function in the beam is in phase (remember back to trigonometric algebra!) with each other and of the same wavelength.
Cooling - This is not temperature exactly. Kelvin(K) is more a measure of energy. While chemists like to go back and forth between (K) and Celsius, Celsius is more a measure of temperature, an average heat of a system. Heat is a form of energy, generally thought of as energy that makes things hot or cold to the touch. Cooling in this context refers to the reduction of overall energy, specifically kinetic energy in the system, reducing K. So, it's getting "colder" because things are slowing down. Basically, for monatomic gases, temperature is the mean kinetic energy (Thanks to unperson for helping me with my lack of articulation). Celsius is not really an appropriate unit of measure here because you can't expect to touch it and expect it to be cold.
Absolute zero - I've discovered that a number of schools are teaching this concept inappropriately. Absolute zero is not the absence of all energy. That's impossible. Rather, it's the equilibrium state (stable and unchanging) in which the system is in the lowest energy state. That is, essentially, the lowest nuclear energy state and all electrons are in their ground states.
On with the cooling! So say you have this gas...how about rubidium (Rb)? I'll admit, I have an affinity for "Ol' Rubi" because that's what we use in the lab. The rubidium atoms in the glass cell are running around all crazy and bumping into each other, lots of kinetic energy, right? Flash them with a tuned laser and they'll slow down. For simplicity, imagine a single Rb atom. It's travelling at some velocity in some direction. Oh, here comes a laser beam from the opposite direction! What will happen? Well, it depends on what the laser frequency is and/or what the atom's velocity is.
Atoms will only absorb a photon if the photon is of the proper energy. The proper energy is the energy required to excite a valence electron in Rb to the next energy level. The energy of a photon is given by E = hf, where h is Planck's Constant, roughly 6.63 x 10^-34 J*s, and f is frequency. If a photon is of this energy, it is said to be at resonance frequency. Rb absorbs photons with a wavelength of about 780 nm. Frequency and wavelength are generally interchangeable by the equation c = f*λ, where λ is wavelength, f is frequency, and c is the speed of light.
We can tune the laser to a specific wavelength by changing the length of the lasing cavity. However we can't just tune it to 780 nm or resonance frequency and expect it to cool! The Rb atom is moving into the laser, and due to Doppler effect, the atoms see the laser at a higher frequency/shorter wavelength (blue-shift). So, we tune the laser a little below resonance, so that when the photons and Rb atom collide, the photons' frequency is blue-shifted up to resonance. You can think of the atom as a bowling ball and the laser as a stream of really fast ping pong balls. The impact of collision/absorption will gradually slow it down.
"But it can't keeping absorbing photons because the energy will skyrocket!" you say. Well, immediately after absorbing the photon, the excited electron jumps back down to a lower level by releasing a photon, spontaneous emission. This can be seen as fluorescence. This doesn't kick the atom off in some other direction because, since emission probability is equal is all directions, the average change in momentum (force = dp/dt!) due to emission in any given atom is zero! As the atom slows down, well, the laser is blue-shifted less and less and eventually is no longer seen at resonance frequency and is no longer absorbed, but you do have a really slowly moving Rb atom!
The next step is to take this idea and perform it in 3 dimensions to do laser trapping!
Sources: My class and lab notes.