Dimorphos is the moon of the asteroid Didymos -- or rather, the smaller pair of the binary asteroid 65803 Didymos (Didymos being the Greek word for twin). Dimorphos was sometimes called Didymos B, and sometimes called Didymoon, until 2020, when it suddenly became important enough to get its own name.
Dimorphos is only about 170 meters in diameter, and orbits the primary at about half a kilometre out. Didymos and Co. jointly orbit the Sun at a distance of 1.0 to 2.3 AU once every two years (actually, 770 days). Every so often it approaches close enough to the Earth to alarm the Earthlings, the last noteworthy approach being in November 2003, at 7.18 million kilometres. But that's not important right now.
In 2018, NASA approved the Double Asteroid Redirection Test (DART), a test in which a spacecraft was to smash into an asteroid. This is a fun idea that other space agencies had looked into doing in the past, but NASA wanted to make the whole thing much more scientific by crashing into an asteroid moon. It is surprisingly hard to measure the delta v of a chunk of rock millions of kilometres away, so by hitting an asteroid orbiting close to a partner asteroid, all they have to do is measure the change in the orbit. Given that the DART spacecraft is only 500 kg (1,100 lb) and will be moving at at 6.6 km/s (4.1 mi/s), hitting an asteroid that weighs about 5 million tons, careful measurements will be important.
The impact is scheduled for October 2, 2022, and will be video-taped for later viewing. While the impact itself should be interesting, the Didymos system is expected to survive the event with minor perturbations, and a proposed follow mission to view the crater -- the European Space Agency's Hera, planned to arrive in 2027 -- is expecting to see very little change in the neighborhood other than that crater. The system's next close approach to Earth, in November of 2123, is not expected to be significantly modified.
The new name was proposed by Kleomenis Tsiganis at the Aristotle University of Thessaloniki, as Dimorphos will (hopefully) become the first celestial body to have its form substantially changed through human intervention, and hence have two forms.