Here's how to type the "extra" German characters on a non-German keyboard, with a few more notes below for Windows users:

Those Windows shortcuts in the middle column usually have to be entered on the numeric keypad, and sometimes it has to be the left Alt key, not the right one. As for the Windows 7 shortcuts, we put "sometimes" because they're unpredictable -- they tend to work only in certain installations of Windows, or in Word but not a web form, etc.

To be honest, the whole process on Windows is so annoying that most users who have to type any serious amount of German text will be better off just switching their keyboard layout. This may interfere with a few other keyboard shortcuts (especially the ones with the right-hand Alt key) but it's usually worth it. Click here for instructions on how to switch your keyboard layout in Windows.

Of the default layouts, the "US-International" one is a good alternative to "German," because it keeps most of the letters and basic punctuation in the same places and uses alt-key combos for the umlaut vowels.