Quit: cancel running or partially typed command.
Abort innermost recursive editing level and cancel the command which invoked it (abort-recursive-edit).
Either quit or abort, whichever makes sense (keyboard-escape-quit).
Abort all recursive editing levels that are currently executing.
Cancel a previously made change in the buffer contents (undo).
There are two ways of canceling commands which are not finished executing: quitting with C-g, and aborting with C-] or M-x top-level. Quitting cancels a partially typed command or one which is already running. Aborting exits a recursive editing level and cancels the command that invoked the recursive edit. (Section 31.26.)
Quitting with C-g is used for getting rid of a partially typed command, or a numeric argument that you don't want. It also stops a running command in the middle in a relatively safe way, so you can use it if you accidentally give a command which takes a long time. In particular, it is safe to quit out of killing; either your text will all still be in the buffer, or it will all be in the kill ring (or maybe both). Quitting an incremental search does special things documented under searching; in general, it may take two successive C-g characters to get out of a search (Section 14.1).
On MS-DOS, the character C-BREAK serves as a quit character like C-g. The reason is that it is not feasible, on MS-DOS, to recognize C-g while a command is running, between interactions with the user. By contrast, it is feasible to recognize C-BREAK at all times. Section E.1.
C-g works by setting the variable quit-flag to t the instant C-g is typed; Emacs Lisp checks this variable frequently and quits if it is non-nil. C-g is only actually executed as a command if you type it while Emacs is waiting for input. In that case, the command it runs is keyboard-quit.
If you quit with C-g a second time before the first C-g is recognized, you activate the "emergency escape" feature and return to the shell. Section 32.9.8.
There may be times when you cannot quit. When Emacs is waiting for the operating system to do something, quitting is impossible unless special pains are taken for the particular system call within Emacs where the waiting occurs. We have done this for the system calls that users are likely to want to quit from, but it's possible you will find another. In one very common case--waiting for file input or output using NFS--Emacs itself knows how to quit, but most NFS implementations simply do not allow user programs to stop waiting for NFS when the NFS server is hung.
Aborting with C-] (abort-recursive-edit) is used to get out of a recursive editing level and cancel the command which invoked it. Quitting with C-g does not do this, and could not do this, because it is used to cancel a partially typed command within the recursive editing level. Both operations are useful. For example, if you are in a recursive edit and type C-u 8 to enter a numeric argument, you can cancel that argument with C-g and remain in the recursive edit.
The command ESC ESC ESC (keyboard-escape-quit) can either quit or abort. This key was defined because ESC is used to "get out" in many PC programs. It can cancel a prefix argument, clear a selected region, or get out of a Query Replace, like C-g. It can get out of the minibuffer or a recursive edit, like C-]. It can also get out of splitting the frame into multiple windows, like C-x 1. One thing it cannot do, however, is stop a command that is running. That's because it executes as an ordinary command, and Emacs doesn't notice it until it is ready for a command.
The command M-x top-level is equivalent to "enough" C-] commands to get you out of all the levels of recursive edits that you are in. C-] gets you out one level at a time, but M-x top-level goes out all levels at once. Both C-] and M-x top-level are like all other commands, and unlike C-g, in that they take effect only when Emacs is ready for a command. C-] is an ordinary key and has its meaning only because of its binding in the keymap. Section 31.26.
C-x u (undo) is not strictly speaking a way of canceling a command, but you can think of it as canceling a command that already finished executing. Section 7.4.