I would like to have some program that would let you execute a command without having a terminal open, like a little dialog box in which you type the command and press OK.

Preferably no dependencies on a particular desktop environment.

In gnome, its easy. I don't know about other desktops but I assume they are very similar. Right click on the desktop and click "Create Launcher". Choose application in terminal and type the command in the command box. You can also create a custom script if you want to run multiple commands or a lot of options, etc. Then create a launcher whose command is your script. That is how I launch skype with my webcam and some nautilus apps. Hope this helps.

You're looking for an application dock or the menus of your window manager in particular. I don't believe there are any that are universally portable, wbar is fairly portable. Avant is pretty popular, but I'm unsure of what it requires library or window manager wise (think it requires a compositing window manager.) Kooldock will run on several wm but requires kde libs and there's a slew of ones that require gnome libs (Cairo, Do-Docky, etc).

WBar is probably your most portable bet as it will run on a non-compositing window manager. Baring WBar I'd guess Avant is the next most portable.

I don't have any desktop environment. Just a window manager.

I'm not looking for application-dock like thing.

I am just looking for a dialog box in which I can type a command (e.g. firefox) and click a button to close the window and run the command.

Maybe it would be easier just to write it myself in PyGTK

I already built the user interface, but I don't know how you would make it launch another process and then exit itself.