One thing to note about applications running on the Mac:
Closing the window by clicking on the red button does NOT close out the application.
Remember that in windows, every window (whether a Windows Explorer file manager window or application window) has the menubar attached to it, which is why clicking on the "X" closes the application as well as the window.
Since the Mac OS has a universal menubar, the application stays open even when you close a window within the application itself. All the red button does is close the window. This is useful in that once you are finished with one document and need to open a new one, you aren't required to relaunch the application to do so as in Windows if you were to close the window.
In Windows, you CAN actually tell the application to "close" the currently open document, but it is more cumbersome than it would be on the Mac. While you would have to go to File-->Quit to let the computer know you are making the effort to leave the application completely, Windows does it "bass ackwards," if you will.
To close out a document you actually have to File and then select Close in order to keep the application running. In Windows, it's too easy to just quit out of the application than it is to close the document within the application. This tends to be a little dodgy especially if you have multiple documents open within that application (meaning that Windows will bug you about saving each document when you didn't intend to close out of the application entirely). Yes, sometimes on the same bar as the menu in Windows there's a close button, but it just makes the interface too cluttered and even more confusing especially to new users that then see TWO "X" buttons on that application (Office is notorious for this).
The best way to find out if you are still running the application is to look at the Dock. If the application is running (even though there's no window indicating so), you'll see that there's a little black triangle right underneath the icon. Also, if you notice on the screenshot you posted, the name of the application is visible on the left. If it were not running, you wouldn't see "Apple System Profiler" listed on the top left next to the Apple menu. It would say "Finder" (which is your Mac desktop) or the name of another application.
If you see the application is open but there's no window, make sure that the name of the intended application is showing at the top left and go to File. There should be options there to give you what you need (Usually something like "New" or something similar).
Here are some screenshots I took of my iMac at work to give you an idea.