Microwaves are electromagnetic radiation (light) that has a frequency between 0.3 Gigahertz (GHz) and 300 GHz. (One Gigahertz is equal to 1 billion Hertz.) This range of frequencies corresponds to wavelengths between a millimeter (one thousandth of a meter) and a meter.
Microwave radiation is often described as being an independent radiation band, but is in fact a subset of radio waves. In fact, in some fields light with wavelengths in the far infrared, microwave and ultra high frequency radio bands are sometimes generically called "microwave" radiation, even though they are technically three separate energy bands.
Microwave Astronomy
In general, Microwave Astronomy falls under the purview of Radio Astronomy, even though the frequencies typically studied are actually in the microwave band.
Water vapor in the atmosphere can interfere with the detection of microwave radiation, absorbing the radiation and preventing it from reaching the Earth's surface. Therefore ground based microwave observatories are usually placed in high, dry locations.
Microwave telescopes are inherently large since the size of a telescope relative to the wavelength that it is attempted to detect will determine the angular resolution (how small an object the instrument can detect). Since microwave radiation makes up the longest wavelength radiation, short of low frequency radio waves, the size of the detector needs to be many times greater than the radiation wavelength (which can be up to a meter) even to resolve an astronomical object the size of our Moon.
Astronomical Sources of Microwave Radiation
The closest source of non-terrestrial microwaves is our Sun. Though the specific wavelengths of microwaves that are primarily emitted by our Sun are absorbed by our atmosphere.
Active galaxies (AGN), powered by supermassive black holes at their cores, produce microwave radiation, and are some of the strongest emitters. Additionally, these black hole engines will create massive jets and lobes that glow brightly in the microwave. These lobes can sometimes be larger than the entire galaxy.
Similarly, the center of our own Milky Way Galaxy is a source of microwave radiation. Likely this is linked to the theorized supermassive black hole at our galaxy's core. While ours is not an Active Galaxy, the center is still quite a dynamic place.
Pulsars (rotating neutron stars) are also strong sources of microwave radiation. These powerful, compact objects are second only to black holes in terms of ultimate density. With powerful magnetic fields and fast rotation rates broad spectrum radiation is produced, with the microwave emission being particularly strong.
In fact, most pulsars are usually referred to as "Radio Pulsars" because of their strong radio emission. (Recently, the Fermi Gamma-ray Space Telescope characterized a new breed of pulsars that appears strongest in gamma-ray instead of the more common radio.)
The Cosmic Microwave Background Radiation (CMB)
When a microwave telescope is pointed in nearly all directions a faint microwave glow is apparent. This is known as the Cosmic Microwave Background (CMB).
Effectively the CMB is the afterglow of the big bang, the event that set our Universe in motion. The residual heat from this event was spread out over the entire Universe, but as the Universe expanded the heat density dropped proportionally. Simply, as the Universe evolves and expands the average temperature has continued to drop.
Today the CMB represents a temperature of about 2.7 kelvin, which manifests itself as microwave radiation. The CMB is uniform across the entire observable Universe (with error). Discovered in 1964 by radio astronomers Arno Penzias and Robert Wilson, a discovery for which they won the Nobel Prize in 1978, it was the first hard evidence in support of the big bang theory.
