“Next Friday” means the first Friday after today. Based on today’s date, that would be Friday the twelfth of October. We could also say (and it would be clearer) “this coming Friday”.
“The next Friday” is not used as often as the expression without the article, and there may be variations in usage that make it ambiguous. To me, it means the Friday following a time that is specified. For example: On September first, we planned to have a party (on) the next Friday (i.e. September 7th). We could also say “(on) the coming Friday.” Or for example: “this Friday (October 12), I have to study, but we can go out the next Friday (October 19th).”
Edits: A couple of typos.