You may run across problems with some applications where they will not quit. When you use the command-Q hotkey an application may neither respond nor shows any error, and if you go directly to the application menu you will see the "Quit" option is greyed out.
This usually happens if there is an application-related notification, such as a warning that the application cannot perform a given function. These notifications may lock out some functions of applications, including the ability to quit them. While the application's Dock icon will usually bounce when these warnings are present, this may not always happen.
If you experience this problem, first try to locate the notification window and cancel it. Click the faulty application to bring it into focus, and the notification window should appear, though sometimes you may have difficulty finding the window.
While notification windows should be kept on top of other windows, errors can have them hidden behind other windows. You can try using Expose to find the window; however, in many cases these notification windows will disappear when Expose is invoked. Nevertheless, toggling Expose may bring them back to the front so you can dismiss them.
If this does not work, try minimizing the windows on the screen to see if that unveils the notification window. Optionally, you can press the Option key, right-click the application's icon in the Dock, and choose "Hide Others" so only the windows pertaining to that one application will be shown.
If you use Spaces, keep in mind that the notification window may be in another space, so even if you have brought the problematic application into focus and have hidden other applications, the window may still be off screen. You should be able to locate the notification window by scrolling through your spaces (like with Expose, it may disappear if you show all spaces at once).
As a last resort, you may need to force-quit the application. Every now and then there may be errors where the notification window is not showing, and yet the application still gets locked with no Quit option as if a notification window is present. If this is the case, you will need to force-quit the program by either using Activity Monitor or choose "Force Quit..." from the Apple menu.