Explain the controversy: Some say Christopher Columbus was a brave explorer and others say he was a cruel murderer. Explain why they say that. What reasons do they give?

With questions like this one, it's always important to remember that the two things—brave explorer and murderer—are not necessarily mutually exclusive. People can do good things and do bad things simultaneously; how Columbus is viewed has changed over time and is a matter of perspective.
There is plenty to support the fact that Columbus was indeed a brave explorer:
1. He was a skilled navigator who travelled across the Atlantic Ocean on four separate occasions at the command of the King of Spain, at a time when this meant journeying into the unknown.
2. Columbus "discovered" the Americas, albeit by accident—he was really heading to India. Regardless, he succeeded in bringing awareness to Europeans of that era of the existence of the Americas.
However, it's important to remember that Columbus was not embarking on these journeys out of the pure intrepid desire to explore. He wanted to reach the Far East so that he could establish a trade passage which would make him a lot of money. There are also many things Columbus did in the Americas which have led to him being viewed controversially, such as:
1. He established a colony in Haiti, without regard for the views of the people already living there.
2. When he came back to Spain to report what he had found, he also brought with him a number of native people whom he had effectively enslaved; they were being held against their will. Columbus continued to enslave natives in the region.
3. The first group of people Columbus encountered, the Taino people, were ultimately wiped out as a result of Columbus's interaction with them. While they survived beyond his lifetime, Spanish groups he brought to the area wiped them out through a combination of deliberate attack, enslavement, and the transmission of European diseases (chiefly smallpox).
4. Columbus's manner of rule over the Indies was considered so tyrannical even at the time that the King of Spain removed him from power. As governor, Columbus tortured and mutilated the native peoples, as well as committing acts of genocide as a means of controlling the population and putting down rebellion.