Like many others, the first few compositions by Brahms that I got to know were the famous Brahms's Lullaby (Weigenlied), the Waltz in A Flat, and the Variations on a Theme by Joseph Haydn, also called the St Anthony Chorale Variations.
Brahms was very fond of the great German composer, Robert Schumann, and his beautiful and accomplished wife, Clara (Wieck) Schumann. Their lives were intensely intertwined, not least because of Schumann's tragic insanity, which left his wife Clara to have to fend for herself and her children without much support, except from Brahms, for a while. This lullaby could easily have its origins in the time Brahms spent with the Schumann children.
The Waltz in A Flat is an elegant stylized waltz, not really long enough to be danced to; in other words, it is a concert waltz for the piano, or the studio, like those of Chopin.
Both these pieces, simple though they appear to be, have little harmonic surprises that are just enough to delight, and after we hear the pieces a couple of times, we can't imagine them being harmonized any differently. The Lullaby, in particular, has the same bass note throughout, repeated gently deep in the bass. The Bryn Terfel performance was a lush orchestral arrangement that obscures the repeated solitary bass note, but once you're aware of it, it can't be ignored.
We're told that the Variations on the St. Anthony Chorale was initially a piano piece, and Brahm's first venture into orchestral composition, and a brilliantly successful one, too. This performance is conducted by Riccardo Muti, with the Philadelphia Orchestra.
Anyone interested in the life of Johannes Brahms can easily find out more. He was brought up in poverty by a single mother, but is considered a good and wonderful man, though his last years were spent in a cloud of bitterness.
Brahms wrote several concertos, all of them among the most popular in their various genres: the amazing Violin Concerto in D, the two Piano Concertos, and a Double Concerto for Violin and Cello. All of these are worth hearing; I'll put in links one of these days.
Finally, for those who want something light and fluffy, Brahms enjoyed gypsy music, and his Hungarian Dances are simply gorgeous: rhythmic and tuneful, intended to be played piano four hands (two pianists playing the same piano), and now orchestrated for full orchestra.