The chances are that you have hardcoded your connection string - included it as a string in your source code - and that the first thing you need to do is change that to use a configuration file of some description.
The simplest is to use the Settings.Settings file:
1) Open your projects Properties in the solution explorer, and double click on "Settings.settings"
2) In the resulting grid, change the Name to "MySetting", and set the Value to "Defaulted value". Leave Type and Scope as "string" and "User" respectively.
3) Save and close the settings window.
4) To read your setting:
string s = Properties.Settings.Default.MySetting;
5) To write your setting:
Properties.Settings.Default.MySetting = "My new setting value";
Properties.Settings.Default.Save();
Then you need to provide a mechanism for changing the configuration file so that you can set the appropriate connection string for the new environment. What that will be we can't say: most development SQL server instances tend to use Windows authorisation for log in, while production ones are generally username and password protected - you will have to find out from the db admin what the string should be.