It seems that there is quite a difference in how the Undo command is handled in various browsers.
In Mozilla and in Opera 7, each textbox has a separate undo buffer. If you start typing into one field and then type into another field, you can undo each separately. Choosing Undo from the Edit menu or pressing the Control+Z shortcut undoes the last change made in the currently focused field. Switching to another field allows one to undo the changes made in that field instead. The address bar and fields in dialog boxes function in a similar manner.
Both Mozilla and Opera provide an Undo menu on the Edit menu, in the context menu for textboxes and Undo is performed when Control+Z is pressed. Both browsers also support the Redo command, which redoes a previous undo operation. Mozilla provides the Redo command on the Edit menu, the context menu, and supports the Control+Y shortcut. Opera doesn't have Redo on the menus but does support the Control+Y shortcut.
Both browsers support multiple steps of undo, which means that choosing to undo several times undoes each successive change before. You can redo multiple times also. This is supported on all fields.
The Windows version of Netscape Communicator supports Undo on textboxes, and supports it separately for each one. Making a change to one field doesn't affect the undo buffer for another. The Undo command is not on the Edit menu, but is available on the context menu and the Control+Z shortcut works. However, only one level of undo is available. Selecting to undo a second time just undoes the undo and goes back to the original state, which is the equivalent of what Redo does on Mozilla and Opera. There is no Redo command and the Control+Y shortcut does nothing in Communicator. I think that this is the normal behaviour of Windows textboxes.
The Linux version of Netscape Communicator is unusual. There is an Undo and a Redo command on the Edit menu, but they are always disabled. Editing a textbox doesn't seem to enable them, which makes me wonder why the commands are there. Netscape Mail 4.x has an Undo command that works in the message area, but no Redo command. There is also an Undo command in the Edit Bookmarks window which can be used to undo the deleting of a bookmark.
The Windows version of IE (5.5 and 6.0) doesn't have an Undo command on the Edit menu, but it is available on the context menu and the Control+Z shortcut works. The Redo command is found neither on the Edit menu nor the context menu, but the Control+Y shortcut can be used to redo.
Unlike the other browsers however, IE only supports one undo buffer per window. If there are two textboxes in a window and text is entered in the first and then in the second, the Undo command reverts the second textbox. Pressing Undo a second time, switches the focus to the first text box and reverts it. The Redo command is similar but in the opposite direction. Since this undo buffer is for the entire window, this will also apply across multiple frames. The behaviour is different from Mozilla and Opera as they handle undo separately for each field.
However, the IE address bar works differently. It has a separate undo buffer that does not support multiple levels of undo. Instead it works more like a Windows textbox (like in Netscape Communicator). Selecting to Undo a second time reverses the undo. There is no redo command for the address field.
Here comes the nasty bit which prompted me to write this piece. In IE, if you have a script which changes the document in any way, either using DOM functions, setting the innerHTML property of any element, or simply by changing the value of a form field (but not by making style changes), the Undo buffer is immediately cleared. Let me state that again: Changing any part of the document in IE using a script clears the undo state so that the user cannot undo any previous changes.. Thus, if you change the document in some way in a key event handler (onkeydown for example), the Undo command becomes pretty much useless. Watch out for this issue if using a textarea where the user might be entering a longer section of text (such as WebMail message or Forum post). Here is a test you can try yourself.
Well that's all the browsers I have available for checking right now. I'd be interested in knowing how various Mac browsers work since I don't have access to a Mac right now.