The WC-135 is at base a modified C-135 Stratolifter, which itself is a descendent of the Boeing 367-80 "Dash-80" prototype. On September 16, 1947, General* Dwight D. Eisenhower commissioned the Constant Phoenix program, under which the U.S. Army Air Force (which two days later became the U.S. Air Force) was responsible for the detection of atomic explosions through the use of atmospheric sampling. The WC-135 is the third aircraft type to be tasked with this mission, and took over the role in 1965 from the WC-50.
The WC-135, like its predecessors, contains several types of equipment for the collection and analysis of air samples. One system maintains a flow of outside air over a filter medium, which is continuously cycled into the airplane and tested for radioactive particles. Another takes regular samples of outside air, compresses them, and maintains them in sealed collection containers for analysis by ground laboratories. It is possible that WC-135 aircraft also contain bhangmeters and other 'realtime' nuclear detonation sensors as well as ambient radiation counters (Geiger counters etc.) for more immediate results.
Currently, the U.S. has two active WC-135 aircraft, a WC-135B and a WC-135W. The latter was instrumental in determining the severity and spread of contamination from the Chernobyl disaster. They remain busy monitoring atmospheric conditions for any sign that any nation has violated the Limited Nuclear Test Ban Treaty of 1963, which prohibited atmospheric testing by its signatories, and, later, The Comprehensive Nuclear Test-ban Treaty of 1996 which prohibits all atomic detonations. These airplanes can be (and have been) deployed to specific areas around the world in times of high likelihood of nuclear events such as North Korea's potential test in 2002 when the aircraft was deployed to Kadena Air Base in Okinawa to make flights around North Korea's surrounding airspace looking for radiation.
The airplane can be and is used for weather research as well; its sampling and measurement equipment is of necessity able to precisely record atmospheric conditions ranging from temperature, humidity, wind speed and direction, and of course the presence of any particulates that can be extracted from or measured in its samples such as pollen, pollution and the like.
The WC-135, like other special reconnaissance aircraft such as the RC-135 Rivet Joint, is maintained by the 82nd Reconnaissance Squadron of Air Combat Command. It is home based at Offutt AFB in southeastern Nebraska.
* - Many thanks to mauler for catching my tryptophan-induced sloppiness and reminding me that in 1947, DDE was General Eisenhower, not President Eisenhower.