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.
The three fifths compromise was a law saying that a slave only counted as three fifths of a person. There are many things wrong with this law, one being that it is immoral. Slaves were not thought of as people in that time, but we now see that this way of thinking was immoral. Second, slaves only counting for three fifths of a person is an inaccurate representation of a states population. The slaves could not vote and had no way to be heard by the government, therefore, states that had slaves were having "property" counting as people who needed to be represented opposed to the states in the north that did not get that. Representation by population was unfair because of this law.
The three-fifths compromise was the result of tension between the northern and the southern states in regards to how slaves were counted in matters such as population count and taxes. A southern slave owner would not want to be taxed for having slaves, but would want slaves to count in terms of population so that the state as a whole would gain more representatives in the House. Northerners would very naturally want the opposite of this to happen. The three-fifths compromise decided that for all purposes, slaves counted as three-fifths of a person. It is clear today what was wrong with this decision. It is absurd nowadays to think of any individual being considered as anything less than one person in the eyes of the law.