If you plan to live in Denmark for an extended period of time, a visa on its own may not cover you for this. You therefore need to be clear whether you are applying for a visa or a residence permit.
EU, EEA and Nordic nationals are able to live and work in Denmark for an unlimited period of time. Other foreign nationals from countries such as the USA, Canada and Australia need a visa which allows them to stay in Denmark for up to three months. Foreign nationals are not allowed to take up paid work during this three month period, and they may have to show a return ticket and provide evidence of sufficient finances to cover their stay in Denmark.
Living in Denmark requires you to obtain a residence permit from the regional government office. In general any stay that lasts longer than three months, or involving studying or working in Denmark, will require a residence and/or work permit. However, there are special rules for EU citizens and citizens from the Nordic Countries as well as Switzerland and Liechtenstein.