How do I make the database central?
By installing SQL Server or similar product on a dedicated computer that will reside within the network your application will be running in.
How do I install this app to Server and client systems?
In this instance, you will probably not have a "Server" application which receives data and sticks it into the database, instead your Client program will directly insert data into the DB and query data from the DB. When you build a solution in Visual Studio, an executable is created and run. In Debug mode, debugging hooks are added to the build so Visual studio can execute the code 1 line at a time and provide information inside the IDE when exceptions are thrown...
You can compile/build a solution into a "Release" format, in which case you will have a slightly smaller .exe (won't have the debugging hooks in it) that can be distributed to your Client machines to execute. (The computer that will run the client must have the proper framework installed or an exception will be thrown immediately when you run the application, and it likely won't say anything about the framework not being installed).
There are solutions inside of VisualStudio that allow you to create installation stuff for your project, I have never used this. Microsoft offers some execelent information about this in the MSDN, I suggest hopping over to it (here are some links for you):
Windows Installer Deployment http://msdn.microsoft.com/en-us/library/206sadcd.aspx
Troubleshooting Windows Installer Deployments http://msdn.microsoft.com/en-us/library/kz0ke5xt.aspx
How do I access the database from the client machines?
Once you have a dedicated computer with SQL Server installed on it, and you have it on the same network that the client machine is on, you can connect to the database using the proper connection information... Your Dedicated SQL Server install will have an IP Address, that is the ServerName. Your SQL Server Install will have databases that you have created, that is the DatabaseName. Your SQL Server install will also have users set up that can access some or all of the databases on it, those are the Username and Passwords you use. And lastly there should be a security setting which blocks incoming connections that are not on a specific port, and that is why you can supply a port with the connection string...
So, the short answer to your last question is, once you have SQL Server installed on a dedicated computer that is on the network the client machine is on, you should be able to directly access that database using a proper connection string:
Need to know the proper format for your connection string? No worries!! Go here - I use this site often because I am a forgetful programmer http://connectionstrings.com/
Hope this helps,