Use little green maximize button in conjunction with a monitor
If you go into Mission Control in System Preferences, you'll find a setting called "Displays have separate spaces". If you enable this setting, you'll be able to run apps in full screen mode on one display, without turning the other display black. There are other side effects that you may or may not like, however. For example, each display will get its own menu bar, and you will no longer be able to position windows such that they lie partly on each display.

Alternatively, if you Option-Click the "little green button" rather than just clicking it, it will often (depending on the application) maximise the window on that display without switching it into full screen mode. This will also prevent the other display from going black, and you may or may not find it preferable to the System Preferences method.