Elerium, otherwise known as element 115, is the energy source for all alien techonologies. Mined from locations beyond human reach, it has a variety of uses. However, it cannot be found on in our solar system...
Elerium cannot be synthesized on Earth; it can only be obtained from UFO Assaults or Alien Base Assaults. Elerium is housed within UFO Power Sources, and is very fragile. If upon starting a UFO assault you find a UFO Power Source to have been destroyed (as a result of a crash landing), its Elerium will surely have been destroyed as well. Be careful about using explosives near intact UFO Power sources, as the Elerium within is easily toasted!
At the end of a successful mission, any Elerium contained within intact UFO Power Sources will be automatically collected and returned to your base. If for some reason you need to end a mission prematurely, you can still collect Elerium by hand, by shooting open a UFO Power Source (generally with a plasma weapon). For more on securing Elerium in this way, see UFO Power Sources.
Perhaps the most efficient way to gain a stock of Elerium is to find an Alien Base and attack the Supply Ships as they land nearby. Leave the base intact to attract more supply ships, as the Elerium is worth more than the score penalty (150 units per intact Supply Ship).
UFO Power Sources can also be found in some Alien Bases, if they have a particular base "module". You can raid such a base without having to win the mission, and have a soldier carry the Elerium back to the entry lift and then abort (see Alien Base Assault#Smash And Grab for more details). Attacking the base's Supply Ships is likely to be a safer and more reliable source of Elerium, however; see UFO Recovery Values#Alien Base Assaults for more details.
On the map, Elerium appears as a purple ball (the same graphic as a Stun Bomb on the ground). This object is worth 50 points of Elerium. During Base Defense, some of your own Elerium from stores may appear on the battlescape. This is a Good Thing if you win the mission, as each unit is represented as a pod and can be recovered for fifty units. This is probably a bug and appears to be absent from the Collector's Edition.
Don't sell Elerium. Although you do have the option to do so, this is to be highly discouraged. It is used to fuel the Firestorm, the Lightning, and the Avenger. It is also used, in varying amounts, to manufacture most Alien Artefacts, Flying Suits, and various other technologies. Given that you can only obtain Elerium in the battlefield, selling it is not a good idea. If you choose to, it will fetch you $5,000 per unit.
Elerium In Reality
Ununpentium, the temporary name for synthesized element 115, was only created briefly in atomic laboratories in 2004 and is not the subject of much research. However UFO 'researcher' Robert Lazar made several claims about E-115 in the 1980's. For example, it would be the only element that had a gravitational field extending past it's own atomic 'shell'. This deviance from the normal rule would be so significant that perhaps the technologies seen in the X-Com Series could become a reality - anti-gravity devices used for both flight and particle acceleration.