So, how does SQL Server Express work, and what is the difference from SQL Server Management Studio?
Only people in my network can access the databases, and my computer/the server must be on and running, correct?
That's a very broad question, it's like asking how does Visual Basic work and what's the difference between VB 2008 Express and VB 2015 pro.

And yes your machine and the server must be running. That would be a neat trick if they didn't.