The major significance of the New York Draft Riots in the Civil War is that they showed the attitudes of many Northerners (especially Irish immigrants) towards blacks and towards the emancipation of slaves.
Many working class Northerners were completely against the idea of freeing the slaves. They worried that the freed slaves would come to the North and compete against them for jobs. Because of this, they were vehemently opposed to the Civil War and especially to the draft. They did not think that it was right that they should have to fight in order to free slaves who would then come and harm their own livelihoods.
So, the major significance of these riots is that they show that the North was not the solid bloc of anti-slavery sentiment that we like to think it was.
Apart from what has been mentioned by Pohnpei, there were other major significant events that occurred soon after the draft riots. The draft riots that were instigated by a section of the public and the media supported by some local leaders were aimed at stopping the emancipation process being fronted by president Abraham Lincoln. These leaders supported the riots for mostly political reasons since power was seen to shift from democrats to republicans and the abolitionists. It was an opportunity to undermine the then union leadership and protect the white citizens from the looming competition. Instead the aftermath of the riots saw the strengthening of the Union League Club and renewed union of New York's elite and the blacks who chose to stay on. This later saw the establishment of a black regiment to support the war efforts. A public parade of the soldiers was done in New York and "In a powerful display, the parade publicly linked blacks with the leaders of the new order being ushered in by the Civil War." These soldiers would go ahead to fight side by side with their white counterparts to free the slaves in the South and herald a new chapter in the nation's history.