There is not a great deal of orchestra repertoire for the euphonium, though they are common in wind ensembles (and especially in brass bands). However, with development of the solo / ensemble combination videos on YouTube, you can find many players experimenting. So, for instance, this individual recording the opening to "Pirates of the Caribbean," or this one. Or, with a recorded ensemble, this example of a rotary euphonium playing "Let it Go."
With all of that said, there are (of course) many other instruments in the brass family and many (many) other pieces. As with strings, it would be difficult to pick any one piece "typical" of the family. Therefore, I will pick a few of my favorites featuring brass alone, along with brass-heavy pieces.
I played a version of the below in college, and absolutely loved it. It's from an arrangement of pieces by Tielman Susato collectively called "The Danserye".
Without knowing the sounds, however, it is difficult to pick out the euphonium. Here is an example of the euphonium parts (particularly around 1:40):
Holst is a composer known for his "standards" for wind ensemble. Two important pieces constantly played by ensembles are his 1st Suite in Eb and 2nd Suite in F, both for military band. The first is here:
More familiar to most people are the movie scores of John Williams and James Horner, often which both feature "big" and prominent brass sections. For example, here is a nice compilation of some of Williams' scores:
James Horner (who recently died) was also a well-known composer of score ranging from Star Trek:II to Avatar to The Rocketeer, and many others. He is not quite as brass heavy as Williams, but he definitely knew how to use the section. For example:
There are, of course, many other movie composers who excel at brass. I'd like to take a quick turn to another composer now, namely Mussorgsky. His work "Pictures at an Exhibition," composed based on artworks by a friend, was originally written for piano. However, many famous composers have orchestrated it, most notably Ravel. Either on piano or orchestrated, the final movement "The Great Gate at Kiev," is stunningly beautiful. On piano, by Evgeny Kissin, is amazing (watch his hands and facial expressions). Orchestrated, played by any number of orchestras in recordings, is simply breathtaking:
Finally, I'd like to highlight two other modern composers, one well known and one less well known. First, the composer Aaron Copland, with a quintessentially "American" sound, and his "Fanfare for the Common Man:"
And second, the composer Alan Hovhaness, and his "Prayer of St. Gregory:"