ControllerMate does this. At work I use the Microsoft Natural ergonomic keyboard and a Logitech mouse. ControllerMate has hardware profiles for both, and I have many of the MS-specific keys remapped to other functions.
Like yours, my keyboard also has an otherwise useless Mail button. With Controllermate open, if I press the Mail button while viewing the Microsoft keyboard profile, the list of available key sequences automatically jumps to the one it just received. You can then program that key to do whatever you like, even when ControllerMate isn't open.
Revisiting this four years later, now with macOS Sierra.
If you have a non-Apple keyboard then you may want to use a more appropriate keyboard mapping. You can select keyboard mappings through the keyboard tab of the System Preferences dialog. Select Input Sources and use the + button to select a mapping appropriate for your keyboard. (You may also need to use the - button to delete the mapping that you do not wish to use before your chosen mapping becomes fully effective.)
If you find the mappings provided by Apple to be unsatisfactory (Apple's British - PC keyboard does not map the pipe to the correct key - adjacent to the left-hand shift key) then you may install third-party layouts or create your own.
Here is a ZIP file containing two alternative layouts files for UK 105 key keyboards. You should unzip the files into /Library/Keyboard Layouts (use sudo).
The new layouts won't be available in the abovementioned Input Sources until you log out and back in. After that, you can select them like any other layout (they may be listed in the others category).
You can edit your chosen layout if it is still unsatisfactory (I chose the British (PC 105 alt) layout but found that Alt+3 emitted cent (¢) instead of the expected Euro (€) symbol). Layout files are simple XML text files. I identified the relevant key code and replaced its unicode output character:
<key code="21" output="¢" />
with
<key code="21" output="€" />
(The change was applied in two places: keymap index 3, for anyOption key (Alt to PC keyboard users), and keymap index 5 for the same with CapsLock on.)
(Once again, a log out/in cycle is required for the change to take effect).
The key code for the 4 key (which, when used with Alt should produce €) is 21. Should you wish to discover the key code for a physical key then you could use the Key Codes app, available free of charge from the Apple App Store.
Should you desire a more GUI way of customising your keyboard then you could look at Ukelele [sic] or Karabiner, formerly known as KeyRemap4MacBook (although the website states that Karabiner doesn't currently work on macOS Sierra at the moment.)
Still unresolved I wanted to identify keycodes without installing anything (think X-Windows xev) and I could not find Apple's standard key layouts (they aren't in /Library/Keyboard Layouts). I guess they are in /System/Library/Keyboard Layouts in a different format. It would be nice to be able to use a standard layout as a starting point for custom edits made as described above.
Best Answer
Your mac for some reason thinks you are using a Japanese JIS hardware keyboard instead of the US ANSI or European ISO. Notice the Yen key at top right. What kind of keyboard do you actually have? This note may help:
http://m10lmac.blogspot.com/2009/12/fixing-keyboard-type-problems.html