To overcome and take control of a place or people by use of military force. The army was able to conquer the city after a long siege.
To successfully overcome a problem or weakness. She managed to conquer her fear of heights by going skydiving.
To gain mastery over or win by effort. He worked hard to conquer the challenges of the new project.
To successfully climb (a mountain or peak), especially for the first time. In 1953, Sir Edmund Hillary and Tenzing Norgay conquered Mount Everest.