It seems you need some basic explanations about something that unfortunately even most programmers don't understand properly.
Files like your HTML page are saved and transmitted over the Internet as a sequence of bytes, but you want them displayed as characters. In order to translate bytes into characters, you need a set of rules called a character encoding. Unfortunately, there are many different character encodings that have historically emerged to handle different languages. Most of them are based on the American ASCII encoding, but as soon as you have characters outside of ASCII such as German umlauts, you need to be very careful about which encoding you use.
The source of your problem is that in order to correctly decode an HTML file, the browser needs to know which encoding to use. You can tell it so in several ways:
So you need to pick one encoding, save the HTML file using that encoding, and make sure that you declare that encoding in at least one of the ways listed above (and if you use more than one make damn sure they agree). As for what encoding to use, Germans usually use ISO/IEC 8859-15, but UTF-8 is a good alternative that can handle any kind of non-ASCII characters at the same time.
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