Install a JRE perhaps? You have one with Processing, but if you don't have one installed on Windows, you cannot run Java programs (and applets).

That's what I wrote: you have a JRE coming with Processing, but it isn't installed. It is just unzipped and Processing know how to find it, but Windows isn't aware of its existence: it is neither in the path nor in the classpath, etc.
On the other hand, when you install Sun's JRE, it sets stuff in registry, creates a CLASSPATH environment variable, changes the PATH, registers to detected Web browsers, etc.
If you don't want to install another JRE, which I can undertand (they are not small!), hit WinKey+Pause, in the System Properties that appear go to Advanced tab, click on Environment variables button at the bottom. On the System variables list, click on PATH and edit it. Add the path to your JRE, where the javaw.exe exists, eg. C:\Java\Processing\java\bin;

Just add the path to your javaw.exe at the end. So it should look like:

%SystemRoot%\system32;%SystemRoot%;%SystemRoot%\System32\Wbem;C:\Program  Files\Fichiers communs\Adobe\AGL;C:\Program Files\QuickTime\QTSystem\;C:\Program Files\ATI Technologies\ATI.ACE\Core-Static;C:\Processing\java\bin

the last path depending, of course, of where you have unzipped Processing.

Maybe there is a (English) language issue here (I am French!).
By "add the path to javaw.exe", I don't mean to add it to javaw, I mean to add to the PATH variable (in the System variables list) the path "of" the file javaw.exe.
Click Edit, go to second field (where you copied your path), hit End key to go to end and type ; and the path (which you can have copied in the Address bar of Windows Explorer).