"Wagner's music is better than it sounds."-Mark Twain
Every negative stereotype about opera can be laid at the feet of Wagner, especially the fat lady wearing blond braids and a viking helmet. He was egotistical, flamboyant, anti-Semitic, adulterous--deeply flawed in many ways.
That said, he was visionary in his concept of Gesamtkunstwerk ("total work of art"), which sought to combine all arts in one--literature, drama, dance, art, and, of course, music. He perfected the use of the leitmotif, a tool in which each character is represented by a musical theme. He built the Bayreuth Festival to house his operas (or "music dramas")--a magnificent structure that could house a huge pit orchestra under the stage, extensive set designs, and strove to give every audience member a good view of the stage.
A lot of Wagner's ideas are incorporated in the modern movie theater--music scores frequently use leitmotifs (Think Darth Vader's theme, or Gollum's theme, or the "Jaws" theme), and theater seating is based on Wagner's design for Bayreuth.
Wagner's love of the epic tale is also legendary. His Der Ring des Nibelungen is a set of four operas, each lasting 3-4 hours long. Speaking of his influence on movies, I think that watching Wagner's Ring Cycle has got to be like watching the entire extended LOTR Trilogy--except all the characters would be singing their lines.
Another thing Wagner did was to change the format of operas, so that instead of there being recitative and arias, where dialogue could move more quickly in the recitative, and arias would end on a cadence, allowing the audience to applaud the singer, Wagner made the music all run together in one long, continuous phrase that never ended until the opera was over--kind of like this run-on sentence.
I sympathize with Mark Twain, and can therefore only recommend Wagner overtures (or preludes)--where you can get a taste of the harmonies, and hear the themes that will represent the characters for the remaining three hours. I do have a soft spot for Tristan und Isolde. His harmonic treatment of the ill-fated couple is poignant and breathtaking.
In the latter half of the 19th century, there was something of a feud between Wagner and Brahms. They had two opposing musical philosophies, and musicians tended to side with one or the other. (If it's not obvious, I side with Brahms.) While Wagner tried to incorporate all art forms into one, Brahms believed that music existed in its own right, and didn't need to have an accompanying story.
Brahms had a gruff personality, and fit the stereotype of a grumpy old bachelor. However, he was extremely good to his friends. He had a close relationship with the Schumanns, and when Robert Schumann was living in an asylum, Brahms lived in an apartment above the Schumanns' home, and helped Clara run the household. Letters show that he was deeply in love with her, but there is no evidence that she loved him as any more than a friend.
As I said before, all composers after Beethoven had inferiority complexes, and Brahms had one of the worst cases. It was made worse when, after the successful premier of one of his own works, he was hailed as the next Beethoven. Feeling unworthy of the title, he was extremely self-critical, and destroyed many of his own compositions.
Some of Brahms' most popular works are his "Hungarian Dances," which I think are an unworthy representation of his genius. One of his most beautiful and inspirational works is his "German Requiem." There's a MoTab recording of it in English. I prefer it in German.