Space Exploration Technologies Corporation, or SpaceX for short, is a private corporation based in Hawthorne, California, United States. They have catapulted to the forefront of the privatization of spaceflight in recent years with their Falcon rocket series, and the Dragon spacecraft. SpaceX's main goal is the perfection of launch vehicles and spacecraft necessary to accomplish not only a manned landing of Mars, but its colonization.
Founded in 2002 by Elon Musk, one of the founders of PayPal and Tesla Motors, it committed itself to creating its own equipment in-house. Their earliest rocket, the Falcon 1, went through a long series of trial-and-error in a series of launch attempts from Kwajalein Atoll in the Marshall Islands, until it finally successfully orbited a satellite on September 28, 2008. It completed its first launch for an outside entity on July 14, 2009, with the launch of RazakSAT, an imaging satellite, for Malaysia.
From there, they moved up to the Falcon 9 rocket, a rocket intended to compete with the workhorses used by NASA: the Lockheed Martin Atlas V and Boeing Delta IV. The first stage of the rocket uses nine Merlin 1 engines (currently Merlin 1D, specifically), and is capable of compensating if one engine fails. This is something only two other launchers have ever been able to do: the Saturn V, and the Space Shuttle. This was demonstrated on a launch in October 2012, when one of the first stage engines exploded. Its Dragon capsule made it to orbit, but NASA refused to allow SpaceX to try to orbit the secondary payloads on the rocket.
The primary purpose of Falcon 9 is the Dragon spacecraft. First launched in December 2010, it is capable of delivering over 7,000 pounds of cargo to the International Space Station. More importantly, with the end of the Space Shuttle program and the limited capacity of the Soyuz spacecraft, it is now the only spacecraft capable of returning bulk cargo to Earth from the ISS. The capsule typically returns by splashing down in the Pacific off the coast of Baja California, Mexico.
The Falcon 9 is also now being used to lift satellites for commercial customers. The first such launch occurred on December 3, 2013, when it lifted a communications satellite for an European customer. The launch vehicle has had over 50 flights since its debut, with flight 19 as its first loss in June, 2015 when it exploded minutes after launch due to an unexpected over-pressurization of the second stage. Another "anomaly" occurred on September 1, 2016 when the rocket spontaneously blew up eight minutes prior to an engine test, destroying the payload and inflicting significant damage to the launch pad. The cause of this second failure was a buckled liner in several of the helium tanks, creating pockets that allowed liquid and/or solid oxygen to accumulate underneath the lining, which was ignited by friction. Following the explosion, SpaceX has switched to performing engine tests only without attached payloads.
Since 2014, SpaceX gives the Falcon 9 first stage the capability to be reused. After making a "soft" water landing tests (in April 2014; the rocket sunk before they could retrieve it) and a few unsuccessful attempts at landing on an ocean-going barge, the evening of 21 December 2015 (the first launch after the loss of flight 19 in June that year), successfully landed the rocket on a landing-pad on dry-land at Cape Canaveral. They finally nailed their first barge landing on April 8, 2016, and successfully performed their second barge-landing in a row just under a month later, with the landing on May 6, 2016 being the more difficult ballistic-reentry trajectory landing, requiring the use of three engines to slow the first-stage on final descent, instead of just one. With this third landing, SpaceX has successfully landed one rocket from each of their 3 reentry trajectories: Return-to-launch-site on the OrbComm 2 launch, vertical reentry on the CRS-8 launch, and ballistic reentry on the JCSAT 14 launch. The ballistic-reentry trajectory is especially important since that's the type of reentry that the majority of Falcon Heavy center cores will need to land from, with the two side-boosters using the return-to-launch-site trajectory to touch down on dry-land.
SpaceX typically releases videos of both successes and spectacular landing failures, to many a space fan's merriment, not just to show the technological challenges of landing a rocket, but because Stuff Blowing Up is also pretty cool. Musk often uses the term "RUD," or "rapid unscheduled disassembly," when such an event occurs.
They are also creating a human-rated version of the cargo Dragon spacecraft. Originally known as "Dragon V2", it's developmental name is "Crew Dragon" when NASA discusses it, probably to avoid an unintended association with the first mass-produced ballistic missile from World War II. Crew Dragon will carry humans to the ISS as well as cargo and is large enough to carry seven astronauts.
As to the company's central objectives, SpaceX has preliminary plans to use Dragon to explore Mars as well, with their heat shield (theoretically) capable of Apollo-style high-speed re-entry.
SpaceX currently conducts launches from Pad 40 at Cape Canaveral Air Force Station, Florida, and Pad 4E at Vandenberg Air Force Base, California. Their mission control center is located at their headquarters in Hawthorne. In December 2013, they entered negotiations to take over the famed Pad 39A at Kennedy Space Center (the pad used by most Apollo and Space Shuttle missions), where they will launch all the Falcon Heavy heavy-lift rockets as well as their manned missions. Their inaugural Pad 39A launch was the CRS-10 mission delivering supplies to the International Space Station on 19 February 2017, which also marked their first daytime landing of a booster at their dry-land landing-zone. They are also building a launch center in Brownsville, Texas, with sights on conducting their commercial payload launches from there.