How to close a window without the application quitting? The OS X menu allows exit, minimize and maximize:

  I know the following shortcuts:

Exit:      command+Q
Minimize:  command + M
Maximize:  ?

When I use command+Q to exit the behavior differs from clicking the red button.  I want the UI to close and the application to remain running.  Instead, the application exits entirely.

How can I accomplish this with a keyboard shortcut?
Typically, an application will offer a way to both close the UI and quit the application. These are usually standardized to command ⌘+W to close the UI/window and command ⌘+Q to quit the application.

Some applications will still quit entirely when you use command ⌘+W, such as Preview. You can usually rely on this shortcut to close the window without quitting, however.

I should also point out that the red button in the top left does not "quit" the application as you mentioned in the question body. It will close the window (just like command ⌘+W).