I want to be able to have the login screen of OSX Yosemite display the list of users and a name and password field. I have seen it done on my school's computers but they were connected to a user server, but I don't have a server. Is this possible? If so, how could I do this? In settings it only allows the selection of one or the other.
As you have discovered, the choices are:
A list of users with their icon
A name and password box
What you see at school is the "other" user - user info is stored on a network server, the complete list is both unreachable and potentially far too long to simply scroll through.
It's certainly possible, just get the "Server" application from the App Store and set up your own machine as a server.