It is true that words such as racist and white supremacists did not exist back there. But both Northerners and Southerners recognized that the Southern Social system was maintained by White Supremacy and that discrimination against black constituted a form of racism, even if they did not use those words to describe it.

One common word was prejudice. Salmon P. Chase, for example, demanded "a manly protest against prejudice against color." While a Republican from a Southern background avowed that "It is not probable, sir, with the prejudices of my early education, that I would be likely to have too great sympathy for negroes." Another said that Democratic propaganda was "an appeal to the lowest prejudices of caste." A Congressman similarly denounced the Democratic opposition as "an appeal to that low, vulgar prejudice which wages war against the negroes because the lowest man in society is always anxious to find some one lower than himself." A Treasury Department official noted that “Prejudice against the colored people is exhibited continually. Prejudice bitter and vulgar.” Black communities themselves believed that "there is a prejudice against us in the South that will take years to get over.”

As shown by these quotes, Northerners recognized that hatred against Black people was a kind of racism, even if the term itself hadn't been coined yet. They also recognized how language could shape the political discourse. Southerners and people who were not bothered by slavery often used words such as "nigger" or "negro" to refer to African-Americans, but more progressive men used more respectful terms, like "colored people" or even "colored brethren." The American Missionary Association dismissed a teacher for “frequently [using] the vulgar phrase ‘niggers’ when speaking of the colored people.” Republicans also widely denounced Southerners as ignorant and prejudiced, even contemptuously calling them "white trash." (a term which is not anachronistic). A planter saw Radical Reconstruction as “long, long struggle against ignorance, prejudice and laziness."

When it comes to White Supremacy, Southerners proudly declared that their system was one where White men were at the top. During the secession crisis, Governor Brown of Georgia successfully appealed to non-slaveholders by telling them that "among us the poor white laborer . . . does not belong to the menial class. The negro is in no sense his equal . . . He belongs to the only true aristocracy, the race of white men.” In his infamous cornerstone speech, Alexander Stephens declared that the Confederacy was founded upon "upon the great truth that the negro is not equal to the white man; that slavery, subordination to the superior race, is his natural and normal condition." Northern Democrats also appealed to White Supremacy. For instance, Stephen A. Douglas told a crowd that the Negro "must always occupy an inferior position." During Presidential Reconstruction, Christopher G. Memminger admitted that Andrew Johnson's policies “held up before us the hope of a ‘white man’s government,’ and … it was natural that we should yield to our old prejudices,” while an Alabamian said that the key to winning control of the state was “appealing to the interests, the pride and prejudices of the white race.”

Nonetheless, talk about race and white supremacy was hampered by the sad fact that many Northerners believed that White Supremacy was a normal and logical system. There were men like the Blairs, who thought "that all within the limits of our Union should be homogeneous in race and of our own blood." But even assiduous defenders of Black civil rights, infatigable anti-slavery men, often expressed some measure of racism. As Eric Foner concludes, "Most Republicans, of course, shared the prevalent prejudices of their times, and were by no means ready to afford complete equality to black citizens." And indeed, Republicans often made a distinction between natural rights (life, liberty, property), civil rights (suing in court, moving wherever they want, testifying), and political rights (voting and running for office). They were willing to grant natural and civil rights to Blacks, but not political rights. It would not be until Reconstruction that most Republicans came around and supported full political rights for African-American. Giddings, for example, said that "We do not say the black man is, or shall be, the equal of the white man; or that he shall vote or hold office, however just such position may be; but we assert that he who murders a black man shall be hanged; that he who robs the black man of his liberty or his property shall be punished like other criminals."

However, and I cannot stress this enough, even if Republicans, including Lincoln, do not conform to our modern perspectives when it comes to race, Republicans were not White Supremacists in their time. The Republican position that Black people are humans entitled to basic civil rights is actually a radical one if you consider that the great majority of Southerners and a significant section of Northerners firmly believed that Blacks were nothing but chatel. Republicans sincerely and steadfastly fought for the civil rights of African-Americans to be recognized, and before, during, and after the Civil War they denounced Southern racism as "prejudice", and sought to build a more egalitarian and just society. Words such as "racist" and "White Supremacy" were not used back then, but everybody had a clear idea that Southerners and some Northerners were racists, and that they fought to uphold White Supremacy, even if they did not use those exact terms.

Sources: Eric Foner's excellent Free Soil, Free Labor, Free Men dedicates a chapter to the issue of race within the Republican ideology. Aside from that, Foner's Reconstruction: America's Unfinished Revolution, McPherson's Battle Cry of Freedom, and Oakes' Freedom National were used as sources for this answer.