How can I swap the Z and Y keys on the keyboard?

This how-to should do the trick for system-wide keyboard remap of single keys: FAQ: How to disable/remap a keyboard key in Linux?

And this how-to is about creating your own custom keymaps: Custom Keyboard Layout in Ubuntu (or just Linux :)

It won't allow you to change it quicky using a shortcut. You'll need some scripting to do so, I guess.

However, from the first revision of your question I assumed you wanted to change layouts. You've revised your question, but here is how to change layouts:

From the tags I assume you're running Kubuntu (KDE) 12.04 changing layouts pretty straightforward:

1. Open 'System Settings'
2. Open 'Input Devices'
3. Select 'Keyboard' on the left vertical tab.
4. Select 'Layouts' on the horizontal tab.
5. Tick the checkbox 'Configure layouts'.
6. Add your preferred keyboard layouts by clickin the 'Add' button and filling in the fields in the dialog. Selecting an English (US) layout will result in 'qwerty'. -- Optionally configure a shortcut key to change layouts quickly.
7. Click apply.

This should cover keyboard layout and behaviour in your user session. For system wide settings including text console, see this question.

It’s very simple. You should edit the German layout file; so open it through a Terminal (Ctrl+Alt+T):

sudo gedit /usr/share/X11/xkb/symbols/de

In the opened file, you can see “z” on the 27th line:

    key <AD06>  { [         z,          Z,    leftarrow,          yen ] };

and “y” on the 38th line:

    key <AB01>  { [         y,          Y,       guillemotright,    U203A   ] };

All you need to do is to switch y’s and z’s. Then save the file and close it.

Log out one time and log in back again to see the result.

I can now select German (qwerty) in the Unity desktop in 14.04:

This gives me exactly what I want.

It is also possible to switch to this particular layout in the terminal:

setxkbmap -layout de -variant qwerty

There is absolutely no need to mess with the system's keyboard layout files!

To switch the z and y keys in your X (graphical) session, execute the following command in a terminal in the session:

a=52;b=29;c=xmodmap;d="$c -e '";$c -pke | sed -nr "s/^(keycode *)$b(.*)/$d\1$a\2'/p;t;s/^(keycode *)$a(.*)/$d\1$b\2'/p" | sh