Use the verb overcome when you talk about someone who has triumphed over adversity. It's great to see a friend who's always struggled in school overcome his learning disabilities and make the honor roll.
You can also be overcome with emotions, or overwhelmed by your feelings. Many people feel so overcome with sadness at a movie — or happiness at a wedding — that they regret not bringing a pile of tissues with them. The verb overcome also means to win or surpass. If you are going to win the marathon, you are going to have to overcome the five runners still ahead of you.