OK, my question is that can i run applications (for example Firefox) without Desktop environment on linux? and if i can't, why?

You can do it.
Use any window manager you like.

More specifically, you will need an X window manager, but a desktop is not necessary. You can google window managers and will get plenty of results.

OK THANKS!!!!

1) What's X Window Manager?
2) Why does it have "X"? Can't it be simple Window Manager?

X11 (or X for short) is the GUI environment in general
it is a reverse client server architecture
the X server on it's own does nothing but provide an environment for graphical programs (clients) to run
a window manager does little other then draw windows around the graphical programs and provide basic menus
a desktop environment is a combination of a window manager with other extra features such as file management etc...
in short you don't even in theory need a window manager to run X programs but that isn't very functional since you wont get title bars or be able to resize the output, but to be functional you need a window manager

X is the set of programs to give you a graphical environment on your PC. If you don't use X you can use the console (black screen with white letters).

There are many nice console apps for mail, news, browsing etc. without using X windows. But they don't have fancy graphics, everything is characters.

If you want to use browsers like Firefox you will have to use the X Window system like they said. You can pick from dozens of window managers. They all give you similar function, the only point is what you like. Some popular ones to try are Fluxbox and xfce (getting bit fatter now, it's really half WM half desktop).

from wikipedia..
The X Window System (commonly X or X11)

Where does 11 come from??

So can i run Fireofx without Window Manager?

As frieza stated, you won't get the title bars or functionality of a graphical environment without a window manager installed, but technically, yes it can be done. In command line mode, type firefox.

Cheers

.... Type 'xinit' to start X with no window manager.

You will get a terminal window : type 'firefox'

ok thanks that's great!

But i don't understand how can you say that There are GNOME applications and KDE applications... And that GNOME uses GTK+ and KDE uses Qt. I don't that's true because GTK+ and Qt are independent of Desktop Environment. They are tookit widgets. and i can use GTK+ or Qt without desktop environment (KDE or GNOME). Aren't I right about this??

Is Window System a part of kernel?

from Wikipedia : "While on some operating systems the distinction between applications, window managers, and their supporting technologies are blurred, strictly speaking, a windowing system does not include windows themselves."

Hmm... how can i run firefox without windows themselves?

1) Yes, GTK+ is the toolkit on which Gnome is based, and the same goes for KDE and QT. I run apps for both under the same DE.

2) No, the window system is not a part of the kernel, that's why we have options.

Cheers

QT and GTK are libraries that are needed to run kde or gnome applications but that doesn't mean you need to be running the kde or gnome desktop to use kde or gnome applications, only the libraries, as such kde or gnome apps can be run in any window manager as long as you have the proper libraries.