First, let's define the three fifths compromise, then place it in context of the time period, and then we will discuss the issue/problem!
Definition: The three fifths compromise was adopted on July 12, 1787. For the purposes of counting population, slaves were only counted as three-fifths of a person.
Context: When the articles of confederation passed (our country's first attempt at a "constitution", if you will), each state was given one vote, regardless of size. The Southern states did not agree with this method, and argued that they deserved more representation in congress because they had a larger population. This was one of the many reasons why the articles of confederation was scrapped in favor of a new system, our Constitution. At the Constitutional Convention, when representation was debated, the Southern states actually wanted to count each slave as a whole person in order to earn more delegates in Congress. The Northern states didn't like the idea of the Southern states gaining so many delegates, so the three fifths compromise was struck - each slave will only count as three fifths a person, as to give the Southerners delegates that reflect higher populations, but a proportion that isn't skewed because the Southerners imported the most slaves to work on plantations.
What's the issue?: First, the moral issue of counting a slave, who we cannot forget is a human being, as only three-fifths the worth of a freedman. Second, representation was being given to a large group of people (considered more as property than human beings) who were not given the right to vote, which begs the question: were they truly being represented? We counted them as bodies to gain more Congressional seats in the South, which put the Southern states on a more equal footing with the North. However, in our nation's founding document, slaves had no rights, so therefore they were not truly represented. There is a glaring logical error there. How can you count them as numbers to determine representation if they will not be represented?
For more information on the Constitution, see the enotes page here.