What exactly did you do that you ended up deleting the forum module? Funnily enough, it's the only module in Moodle that cannot be deleted from the Administration » Configuration » Modules page. Do you mean you deleted the \mod\forum directory? If so, you should get the forum folder files which match the moodle version you have on your server. Unless you also deleted the forum database tables, in which case you'd have to reinstall.
What happened when you "downloaded the latest forum module of this site and copied its contents into the old forum folder"?