Who organized the rally?
tl;dr: The rally was organized and promoted by Nazis and white supremacists. Some less extreme elements of the far right did not attend because of the negative imagery.
The rally was officially registered by Jason Kessler, a previously unknown white nationalist engaged in racist "white genocide" and antisemitic ""cultural marxism" conspiracy theories.
Wikipedia says that the rally was organized and promoted by these groups from the far-right:
Among the far-right groups engaged in organizing the march were the clubs of the neo-Nazi website The Daily Stormer, the neo-Confederate League of the South, the National Policy Institute, and the National Socialist Movement. Other groups involved in the rally were the Ku Klux Klan, the Fraternal Order of Alt-Knights, the 3 Percenters, the Traditionalist Workers Party, Identity Evropa, the Oath Keepers, Vanguard America, the American Guard, the Pennsylvania Light Foot Militia, the New York Light Foot Militia, the Virginia Minutemen Militia, the Nationalist Front, the Rise Above Movement, True Cascadia, and Anti-Communist Action.
Those are all Nazi or white supremacist groups.
Speakers at the rally were among other Jason Kessler, Richard Spencer, Matthew Heimbach, or Mike Peinovich. All of them are white supremacists.
The SPLC says that:
“Unite the Right” is expected to draw a broad spectrum of far-right extremist groups – from immigration foes to anti-Semitic bigots, neo-Confederates, Proud Boys, Patriot and militia types, outlaw bikers, swastika-wearing neo-Nazis, white nationalists and Ku Klux Klan members – all of whom seem emboldened by the Trump presidency.
The ADL agrees:
A variety of white supremacists, self-identified white nationalists and others on the extreme right are gearing up for the “Unite the Right” rally on August 12 in Charlottesville, Virginia. The event has the potential to be the largest public gathering of white supremacists in at least a decade
Gavin McInnes - known for his right-wing politics and antisemitism - declined to attend because he did not want to be associated with Nazis.
The occidental dissident stated beforehand that the "alt-lite" will not attend because of the level of extremism:
The Alt-Right will be there. Southern Nationalists will be there. White Nationalists will be there. The Alt-Lite was invited to come, but predictably backed out over race.
The far-right Proud Boys denounced the rally because of white supremacists:
The imagery looks like something that comes from 1930 Germany. Where are your American flags? [...] Just look at this flyer and try to tell me it doesn’t look like Nazi propaganda [...] The truth is, we don’t need white supremacists in Proud Boys
What was the rally about?
The initial reason for the rally was to protest the possible future removal of a confederate statue.
However, neither the promotion of the event nor slogans and signs at the event focused on this.
The flyer from the Daily Stormer can still be seen on google images. It depicts a Nazi destroying a Star of David with the tagline "To end Jewish influence in America". It does not mention the statue.
Slogans during the rally were for example "You will not replace us", "Jews will not replace us" and "blood and soil". Hitler salutes were shown frequently, and people marched alongside Nazi flags and swastikas.