According to merriam-webster.com, a controversy is "a discussion marked especially by the expression of opposing views". It is a dispute -- a disagreement.
The word comes from the Latin contro, meaning against, and vertere, meaning to turn. To turn against.
Why do issues become controversial?
In societies, issues become controversial if they represent a problem that is hard to solve. Some controversies boil down to an argument about how to allocate resources, like time, or money, in order to get a solution. If resources are scarce, people will argue about how or whether to spend them to fix the problem. For example, switching away from fossil fuels to slow climate change is a huge controversy. Some people say doing that will cost too much money. Others argue we'll pay more to fix things later -- or we won't be able to fix it at all, so the time to spend is now.
Other problems are less financial and more social in nature. Here people will argue based on what they believe is proper behavior, leading to discussions about whether society should make laws that ban certain behaviors, or that punish people for doing them. An example is physician-assisted suicide. Some people believe this should be against the law because they believe it is wrong for a person to enlist a doctor's help to end their lives. Others see this as a personal decision that should not be dictated by law.