There were four main ways African-Americans were denied their 15th amendment right to vote in the late 1800s, all used to a different extent depending on which part of the South you were talking about at any given time.
The first was the Grandfather Clause, which simply stated that if your grandfather had not had the right to vote, then neither did you. This essentially meant that almost no African-American could vote. While obviously unconstitutional, it was not challenged in court and overturned for some time.
The second method involved a literacy test. This was the most widely used system of disenfranchising blacks. The test would be given by a white voting official, and they alone would get to determine if the person was literate enough to vote. Of course, whites and blacks were not tested equally.
Third, a poll tax, or a charge to vote, was used in many places. Often times a few days wages for a free black, they would have to choose between paying money they really didn't have to spend or not voting. Even if they did pay the tax, they would only have white candidates to choose from, so it usually kept them from voting. The poll tax finally had to be abolished through Constitutional amendment (the 24th).
Lastly was plain old intimidation. The Ku Klux Klan, while mostly underground and disbanded in the late 1800s (it would rise again in the 20th century) was the social police force, and white society often let it be known that any black man who tried to vote would "get a reminder not to" later.
For many blacks during this difficult time, voting was not worth the expense or risk to their personal safety. They would not vote in large numbers until the 1960s.