How can I run my application (written in C#) and use a SQL Server database without having to install SQL Server first?
If it needs to connect to a Microsoft SQL Server database in order to work, then you need to run it on a network where you have a Microsoft SQL Server installed somewhere.
You can use the SQL Server Express, found here, if you haven't gotten the full server available.
You might use the SQL Server 2008 Compact Edition. This will include a small version of the SQL Server as a DLL directly into your application. It has by far not all features of a stand alone server, but is often sufficent for simpler tasks.
If installing a server is not a problem, just get SQL Server 2008 Express Edition. It is free, too, and somewhere between the power of the Comapct Edition and the comercial editions.
If the server is just running on a other machine, just change the connection string.
If your question is about running a database application without a database, which would be kind of odd, then you need to gracefully handle exceptions from the connection, and have an alternative execution path that does not require data, or gets data from somewhere else.
If your question is about running a database application without a LOCAL database, then that's easy, just point your data source to a remote database.
If your question is about something else, then I am totally confused.