Much like the pharmacist's sign which reads "We dispense with accuracy," the term "modern music" is a rather ambiguous phrase. However, as far as the Dallas Festival of Modern Music is concerned, the term is used broadly to refer to music of the 20th and 21st centuries. The festival's primary area of focus tends to be on art music, or what might more colloquially be called "modern classical music." However, even that focus leaves an almost unimaginably huge range of styles, genres, and musical movements to be considered.
Just as the pace of change in 20th century literature, art, and technology dwarfed the pace of previous centuries, so did music see extremely rapid development. Take any 50 year span from the past; let's say, 1750-1800. This period saw us moving from the world of J.S. Bach to that of Haydn, Mozart, and early Beethoven. While even a casual listener can discern significant stylistic differences, the basic ideas are generally very similar. Now, let's move forward and similarly examine 1900-1950. The period begins with the progressive, yet tonal, extended harmony of Gustav Mahler and Richard Strauss and ends with John Cage, the composer most famous for his work 4' 33", quite literally four and a half minutes of silence!
While the eclecticism of 20th and 21st century music may initially seem overwhelming, the great thing is that there is absolutely something for everyone! A great point of departure is with the French Impressionists of the early 20th century. Consider these works by Claude Debussy and Maurice Ravel.