That particular election was very contentious. Without getting into a lot of details, a close modern analogy would be the Bush/Gore race from a few years back. Eventually, Republican Rutherford B. Hayes was awarded the presidency; however, it's generally believed that it was on the condition of ending the Reconstruction in the South, which was a Democratic goal. While Nast was an abolistionist and often depicted Black people with a level of dignity not afforded them almost anywhere else at the time, he was also a strong supporter of Reconstruction and was upset at its ending. He blamed ignorant voters for making the election so contentious that the compromise needed to be struck. And, in his eyes, ignorant voters were recently freed slaves and Irish-Americans, both caricatured here.
Over at this wikispaces page they summarize the comic this way...
By setting the Irish and Blacks as equal on the scale, he is asserting that their votes are equally inferior. On the upper portion of the scale, “North” is inscribed on the side with the Irishmen, and “South” is inscribed on the side with the Black man. This suggests that Nast believes that North and South are equally negatively affecting United States’ politics. The controversy regarding the Election of 1876 was not just the Black man’s fault; the responsibility is equally shared by Black and White. In both instances, there are outside forces affecting their voting decision other than their personal political beliefs. For the Irish, it is the Roman Catholic Church; for the blacks, it is the white people that they depend on for their livelihood. Both the Church and the white Southerners generally supported the Democratic party; whereas Nast and Harper’s Weekly subscribed to Republican doctrine.The Irish were racially stereotyped very poorly for generations. They were considered lazy, perpetually drunken louts, not very far removed from apes. In that respect, they were viewed similarly to Black people but, by virtue of their skin color, they just weren't quite as bad. Absurd as it seems to me, I've seen old references where someone makes the comparison, verbally or visually, of apes evolving to Africans evolving to Irish evolving to Caucasians.
Interestingly, while Nast did have a prior history depicting Blacks in America with a level of dignity, rarely resorting to even the generally accepted visual tropes of the day, he evidently had few qualms portraying the Irish in a negative light. He regularly showed them as drunken neanderthals, and here is no exeception. A German-born immigrant himself, one wonders if Nast felt more resentment against the Irish because he was in more direct competition with them. Allegedly, when he was a child in New York City, he was frequently bullied for his small size and he may have trasnferred that general resentment of a handful of local Irish boys to the Irish as a whole.
Despite being perhaps the most widely recognized American cartoonist at the time, Nast's career began going downhill not long afterwards. His importance diminished significantly after he left Harper's Weekly in the 1880s (and, ironically, Harper's Weekly's significance declined without Nast) and Nast lost most of his wealth in 1884. While he continued working in a variety of (mostly) artistic capacities, he experienced a number of commercial failures but was eventually given a consulship by President Teddy Roosevelt, largely due to his work from years earlier.