> I have a problem that I cannot solve and would appreciate some help. 
> From what I understand about variables in XSLTs, you can't 
> change the value. So if that's true, how can I increment a counter?

You need to think about a higher-level description of your problem, and not
about how you would solve it in an imperative programming language.
> 
> 
> In my scripting experience, I would sort the list by 
> category, create a counter and loop through the list of items 
> comparing each subsequent item to the previous item. It it 
> was the same, I would ignore the category. That may not be 
> the most elegant solution, but it works

This "sorting and comparing to the previous category" is a procedural
algorithm for doing grouping. Grouping in XSLT 2.0 is done using
xsl:for-each-group; in 1.0 it's done (clumsily) using keys as described at
http://www.jenitennison.com/xslt/grouping.

Numbering items within a group can be done using the position() function.

Michael Kay
http://www.saxonica.com/