I have some simple programs on my web site, for download :-
www.engineeringspreadsheets.co.uk, see Sling Design Program

When a user presses the link 'Free Demo Version' to download a program, how can I provide some sort of counter, which tells me how many times the program has been downloaded ?

divert to a page that counts the hits, then start the download.

Yup, thats the way to do it. PHP and mysql might be the way to go for this (at least i would choose it) I have a working script for this if you would like it, i dont have time to post it up now, bt if you pm me or post then i can send u it when i do.

Stoot

Why not do this:
When the users click "download" a popup window containing a hit-counter loads via JS, then automatically closes in about 2 seconds, the page under it is then forwarded to the actual download site.