RICHARD WAGNER

Wagner was a revolutionary. Few artists have pursued a vision with such uncompromising intensity. I would argue that along Bach and Beethoven, and perhaps Schoenberg, Wagner is one of the most influential composers of Western Music.
Of course, one can hate Wagner, and particularly hate his beliefs or how he was later exploited, but it would be wrong to deny his importance. His music is best characterized as paroxistic, of unprecedented scale, exploring tone, mood and feeling with frenetic power and unrelenting intensity. Because his musical dramas provide such a broad canvas, Wagner's music takes the listener to an enthralling obsessive world, with its own artistic rules and unique depth of expression.
Wagner not only revolutionized opera, but music as a whole. To sustain the scale of his opera works, he invented new structures and concepts, which remain hugely influential to this day. Wagner set the tone for the Post-Romantics, and opened the door to 20th Century Music.
Wagner did away with the constructions that had carried opera (and music more generally) to invent his own vocabulary, his own rules. His dramas strive to be a Musical Continuum, a profound unity that binds Wagner's gigantic works (15 hours of music, in the case of the Ring!), which wasn't achievable with forms available to him. Beyond music, Wagner envisioned his work as Total Art, calling on to all disciplines of Art to fulfill a single vision. Wagner not only wrote his own texts, but went so far as building the Bayreuth Theater to accomodate his productions!

In his musical revolution, Wagner did away with structured melody, and pushed harmony to the verge of atonality.

Leitmotivs --
Wagner's musical continuum is achieved largerly through the systematic use of Leitmotivs. This allows him to do away with traditional melodic or form development, and not compose his operas as collections of musical episodes that don't necessarily relate together. Each Opera (or series of Operas in the case of the Ring) is based on a catalog of short musical themes, or Leitmotivs. This becomes Wagner's vocabulary, the small building blocks out of which he assembles his gigantic architectures.
Leitmotivs are generally short, and can be comprised of any of the characteristic elements of music: Melody, Rhythm, Harmony, or even Orchestral Color. While the melodic motives are easier to comprehend, they can be as simple as a rhythm (for example the Schmieden Motiv in the Ring), two harmonic chords (for example the towering Power Motiv in the Ring), or be associated with orchestral color. Most are small enough that they can be used as building blocks, intertwined together in complex orchestral textures, often combined together, etc. The motivs have names, and are generally associated with a person, object, place or concept, but their appearance in the music is far from simplistic. They're presented in an intricate web, which often brings new significance to the text or the action.
This leads to a lengthy but fascinating discovery process as one grows familiar with Wagner's works. The intricate play of the Leitmotivs, their uncanny complexity, slowly become evident, as a picture gradually coming into focus, until one realizes that nearly all the music is somehow based on these architectural elements. There are very few moments in Wagner Operas that aren't derived from the Motivs.
Which is why there is no (or shouldn't be) "Great Airs" from Wagner's Operas. The work exists as a whole, from which it's difficult to extract a piece and have it stand on its own. Also, the Leitmotivs are almost always played by the orchestra, not the singers. The orchestra carries the structure of the music, with the singers "on top" of it, the opposite of an orchestral accompaniment.
Wagner didn't invent the Leitmotiv of course. In fact the idea was popular with the romantics (for example the Idee Fixe that repeatedly haunts Berlioz's Symphonie Fantastique.) But the magnitude at which it is used by Wagner, and the fact that it becomes the underpinnings of such large structures, was unprecedented.
The use of the Leitmotiv provides to the listener a discrete structure that becomes apparent only after repeated auditions. Once these themes emerge, their expressive power is staggering. Wagner's works produce a haunting obsessive feeling, as one starts to grasp their immense complexity. A subtle allusion to a motiv, buried in the rich orchestral texture, can create unbelievable nostalgia, or the precipitated overwhelming accumulation of motivs can lead to the most frenetic intensity. The possibilities are endless...

Chromatism --
In addition to doing away with traditional melodic development, Wagner also questioned the foundations of tonal music. His music is largely based on Chromatism, where the music's tonality is perpetually shifting, to the point of being lost. Wagner was not alone in this (in fact "chromatic" passages are common in Bach or Beethoven), but with Franz Liszt he pushed the technique to become an unmistakable element of style. (This was later embraced by the Post-Romantics, and led to Schoenberg's repudiation of tonality in the early 20th Century.) Parts of Wagner's music are built on such a shifting tonal structure that the music can no longer be analyzed tonally (most famously, the Prelude to Tristan und Isolde.)
Wagner's ever shifting tonal landscape, most evident in Tristan, creates permanent tension, often a feeling of longing, as if the music was perpetually seeking unattainable resolution.

The scale of Wagner's works is particularly remarkable for the Romantic period. The Romantics generally did not produce successful large-scale works. This was due to their taste for fantasy over form, which produced very imaginative music, but was rarely able to carry through larger formats. There are of course notorious exceptions to this (e.g. Liszt's B Minor Sonata). And sometimes the collections of fantasy pieces so favored by the Romantics stand well as a whole (e.g. Schumann's Kreisleriana, Chopin's Preludes...). But these feel almost like accidents, and certainly don't hold together as a Beethoven Sonata, for example.
In the case of Wagner of course, a large part of the structure is provided by the libretto, but the music does stand on its own (I've seen fantastic concert performances of Acts from his Operas which I've actually preferred to stage performances, much like the Rite of Spring plays so well in concert).

Der Ring des Nibelungen:
Rheingold (prologue)
Walkure (day 1)
Siegfried (day 2)
Gotterdammerung (day 3)

Tristan

Parsifal