utf8 or latin1 encoding – German

Short answer: If your editor can use Unicode (and in my opinion any good editor should save files as UTF-8 by default), then use Unicode.

The main advantage of Unicode (and its main encoding UTF-8) is that it makes text files universally readable: Someone writing German umlauts can use the same encoding as someone writing Korean. Prior to that, when obtaining a a text file, there was no way to know which character encoding to apply to it: Latin 1? or Latin 9? or maybe a Russian encoding?

Also for documents that use more than 256 different characters, UTF-8 is very useful. With it you can for example use both "ü" and "ł" in the same text (as I do here).

Both modern (experimental) TeX compilers (LuaTeX and XeTeX) use Unicode/UTF-8 by default as do modern programming languages (i.e. .NET) and operating systems.

If you can: do not use \usepackage[utf8]{inputenc} nor \usepackage[latin1]{inputenc}. Use LuaTeX:

\documentclass{article}
\usepackage{luaotfload}
\usepackage[EU2]{fontenc}
\usepackage{lmodern}
\begin{document}
Das Mädchen ging über die \textbf{Brücke} nach \textit{draußen}.
\end{document}

This will give you access to all modern things (OpenType fonts for example) while keeping most of the backward compatibility.

Wait for TeXlive 2010 (or get the pretest) and you have a decent environment for LuaTeX. Million thanks to the few people who make the LuaLaTeX packages!

If you are able to read german: see the site http://www.luatex.org for more examples (especially on fontspec).

I'm of the idea that, whenever possible, you should stick to utf8 (unicode) as it is finally the solution about character encodings on computer documents. Of course you only have to make sure that your text editor supports and behaves nicely with unicode documents.

In either case, if you have to work with collaborators, you have to make sure that they do also use decent text editors that can handle the appropriate encoding without messing with the special characters. For documents with many collaborators I've learned the hard way that it is better to stick to ascii and use (La)TeX commands to produce special characters.