Mount Everest was first climbed by Sir Edmund Hillary and sherpa Tenzing Norgay. The two climbers agreed upon not revealing who was first among them, and to this day noone else knows. Update: I've found a reference that mentions that Tenzing revealed in a biography before his death that Hillary was first and that they didn't think much about it.
Named 1865 after Sir George Everest who was the leader of the Great Trigonometrical Survey in 1852, when the westerners found the mountain, then referred to as Peak XV. It was then measured to be 29,002 feet, which is less than 30 feet off today's value. Pretty good don't you think!
Tibetan name: Chomolungma or Jonolungma, meaning "Goddess, mother of the world"
Nepali name: Sagarmatha, meaning "Goddess of the sky"
Chinese name: "Chu-mu-lung-ma feng" (thx liveforever)
The perhaps most famous climbers that never made it, were George Mallory and Andrew Irvine, who disappeared very high up on the mountain in 1924. Mallory was one of the pioneers in Everest climbing. His body was found in 1999 by an expedition which specifically were looking for the two British climbers. It is not known whether Mallory and Irvine actually made it to the top - 30 years before Hillary and Norgay. It is believed however that they never reached the summit, but in fact died at 8200 meters - where Mallory's body was found - without having been higher up on the mountain.
There are three ridges that lead to the top of Mount Everest, and consequently the mountain has three faces. Successful ascents have been made from all ridges and faces. The mostly used path is the Southeast Ridge, which was the one Hillary and Norgay used, and it's the route most-used by commercial expeditions. This is reached by climbing the Khumbhu Icefall and then passing the under Southwest Face. This leads up to the South Col, which is a plateau at 7925m, 26,000 feet, where the Southeast Ridge begins. This route lies entirely in Nepal, and intrepid would-be climbers must pay for a climbing permit before embarking. These permits are sold by the Nepali government, and they normally go for approximately $45,000 (USD) apiece. In the early 2000s, maiost rebels in the areas along the treks (usually from Lukla or Jiri, both of which are about thirty minutes' flight from Kathmandu) to base camp have added an element of danger to those on their way to base camp.
It is also possible to reach the South Col via the East Face, the Kanshung Face, which lies in China. The older expeditions tried, for political reasons, to climb via the West Ridge, on the border of Nepal and China. It was on this route that Mallory and Irvine disappeared. Nowadays, climbing from the West Ridge in Tibet is not allowed by the government of China, and climbing the Kanshung Face is usually not allowed to foreigners except under special circumstances.
The third ridge, the Northeast Ridge, is mostly used in combination with the North Face. This was the route chosen by Reinhold Messner in his first solo ascent without oxygen.
Expeditions, usually of the non-commercial variety, have also climbed the relatively unused Southwest Face to the summit.
Other notable records:
Facts from different sources such as: everest.com, britannica.com and various climbing websites.