You don't need SQL Server Management Studio on the machine but you need SQL Server. Management Studio is just the user interface to manage SQL Server.
Second, you don't need SQL Server on all the machines if you your application is connected to the SQL Server via network (intranet/internet) and you can just communicate to one SQL Server.
I think the second is the one you probably want as well otherwise it does not make sense to have SQL Server at all for every single instance of application.
Look at the MSDN documentation to understand how you can use Connection String in config files:
http://msdn.microsoft.com/en-us/library/ms254494(v=vs.110).aspx[
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