connect your app to database through switch

In the connectionStrings section in your web\app config you specify the different connection strings:-

<connectionstrings>
  <add name="ConnString1" connectionstring="Data Source=serverinstancename;Initial Catalog=DBName;User Id=username;Password=password;" />
  <add name="ConnString2" connectionstring="Data Source=serverinstancename;Initial Catalog=DBName;User Id=username;Password=password;" />
</connectionstrings>

then in your code you could have a method which returns an SQLConnection based on the criteria in your switch

public SqlConnection getConnection()
{
   string conn = "";
   switch(yourDependancy)
   {
   case a:
       conn = configurationManager.ConnectionStrings["ConnString1"].ConnectionString;
       break;
   case b:
       conn = configurationManager.ConnectionStrings["ConnString2"].ConnectionString;
       break;
   }
   return new SqlConnection(conn);
}

you will obviously need to check that values exist\nulls etc and maybe include a default case for your switch but this is one way of doing what I think you are after.

Where is the problem? Replace "localhost" in your connection string with the name of the computer the database is running on.
Depending on the database engine (which one - why don't you tell us?) you may have to configure it to accept connections from different machines. The firewalls should be configured to allow the communication.