If you're just gonna listen to one: http://www.youtube.com/watch?v=doeFtpj83fk
Background
Dvořák was born in Bohemia, and is considered one of the most important composers of the Romantic period. He wrote choral, chamber, and symphonic works, as well as concerti and operas. He was part of the nationalistic movement in the later part of the 19th century, and included lots of Czech folk music in his compositions...
Orchestral Music
Dvořák spent 3 years in the US (1892-1895), during which he composed two of his most well-known and respected works...
New World Symphony: http://www.youtube.com/watch?v=ETNoPqYAIPI
Cello Concerto (Rostropovich): http://www.youtube.com/watch?v=xxYbF-Yzdf0&feature=results_video&playnext=1&list=PL119A9750AB63AA9C
One of the most recognizable pieces by Dvořák is probably his Slavonic Dance #2: http://www.youtube.com/watch?v=doeFtpj83fk
[Also beautiful... Serenade for Strings: http://www.youtube.com/watch?v=Ah14jOLKnSs]
Chamber Music
Dvořák wrote some amazing chamber music, including string quartets and quintets, and pieces including piano as well. My favorite (a piece I was truthfully obsessed with for a while) is his second Piano Quintet.
Piano Quintet, op 81: http://www.youtube.com/watch?v=RH8vwG3iFe8 [I really love this recording of a live performance by Natalie Erlich and the Borromeo String Quartet. It's hard to navigate the five YouTube clips that make up this work (and I wish there were a playlist of this piece) but it's worth it.]
Other Music
Another piece many people have probably heard is from his Opus 101, which consists of 8 humoresques. The most famous is number 7, which is usually just called Humoresque, even though it's part of a larger publication... It's been arranged for many different instruments and combinations, but is often heard played by a violin:
Humoresque -- piano: http://www.youtube.com/watch?v=WmAZoexenx8
Humoresque -- violin: http://www.youtube.com/watch?v=uB8mzdO3MnI