The English King Henry VIII desired to divorce Catherine of Aragon to marry Anne Boleyn. However, Catholic teaching forbids divorce, so Henry had to receive special permission from Pope Clement VII. Aside from Catholic doctrine, Clement refused to grant Henry's request for two main reasons: it would reduce papal authority and it would offend Emperor Charles V.
Before he could marry Catherine, Henry had to receive special permission from the Pope (Catherine had briefly been married to Henry's brother, who died soon after the wedding. The Pope agreed to annul this marriage so Henry could wed Catherine). Henry asked Clement VII to declare his marriage to Catherine invalid, which would directly contradict the declaration which had allowed him to marry her originally. The result of this action would have been to weaken the authority of papal decrees.
Additionally, Catherine was the aunt of Charles V, the Emperor of the Holy Roman Empire. Charles' troops mutinied and sacked Rome in 1527--taking Clement VII prisoner in the process. Charles was embarrassed by this act, but it clearly showed the state had the upper hand if conflict arose with the Catholic Church. If Clement granted Henry's appeal, he would have incurred Charles' anger, and whom he clearly was in no position to openly oppose.