This article was written by Daniel Schlichtholz, the developer of MySQLDumper - a free and useful script for backing up and restoring databases. The original language was German and I have translated it to English, because there are people using umlauts and other special characters in other languages who would benefit from reading this article.
The original article, in German, can be found here.

So, here it goes.

The Umlaut-Problem:

There are lots of forums with heaps of reports of umlauts being displayed erroneously.
Attempts to help come from many sides, but hardly anyone seems to have a really complete picture of the problem.
Even web hosts‘ support hotlines seem to reach their limit of knowledge with this one.
There are incredible amounts of half-truths. They prove correct in certain circumstances, but in the end they don’t usually help people because their setup may be different.

In my role as developer of MySQLDumper I naturally spent a lot of time evaluating this situation. And I now believe to be able to offer a complete overview which covers the different aspects of this problem.
A lot, and I mean a lot of research has gone into the explanations in this article.

Be forewarned though: The matter is not simple and cannot be explained in 2 minutes!

There’s a reason why the cries for help in many forums are piling up while hardly anyone can really help.
If you’re looking for a simple explanation („Where do I click to make it work? “), don’t bother reading any further.
This article is long and requires receptivity. What you get in return is, understanding the problem thoroughly.