The Euphonium is a brass instrument with a conical bell. But more importantly, this instrument just rocks! It has the range of a trombone with better technical capability (it has valves). Most people have never heard of a Euphonium. The odds are, though, that you've seen one before but had no idea what it's name was. (Or you mistakenly thought it was a sawed off Tuba)
As a member of the low brass, it plays with a large sound, and has a range similar to that of a trombone. It's valves allow for fast technical passages, (sometimes even doubling the woodwind parts!) and its bore and shape allow for a deep, rich tone in slower solo passages. It is truly a versatile instrument and a lot of fun to play and listen to.
The Euphonium got its name for having a euphonious tone. I just think its plain cool. The one flaw orchestras have is that they don't have any Euphoniums in them. It is primarily a band instrument. The best bands have multiple Euphoinums in them. They ususally sit with the trombones and tubas in the back of the band.
Almost as common as people not knowing what a Euphonium is is people confusing a Euphonium with a Baritone. The key difference between them is the bore of the instrument. The Euphonium has a conical bore, whereas the baritone does not. This lets the Euphonium play with a darker, richer sound than a baritone. This is similar to the difference between a cornet and a trumpet. Many "baritones" that people play in the US are merely Euphoniums with front facing bells. An actual baritone is smaller and has a very distinctive look. Euphoniums usually have four valves.