tap talk is an app, I think for mobile users. I have it on my iphone and ipad, so that I can access this forum. it was pretty easy for me, and i'm not that technical.
Tapatalk is an app that has been around for ages. It aims to make older/more legacy forums more mobile friendly and also of course to attract more end users to the tapatalk eco system and various in-app purchases.
There's a server component which the forum admin can install and of course the app on the mobile device. Users visiting your forums on a mobile device will be prompted to install the app every time they visit.
They make their money via in app purchases as well as custom branded tapatalk UX for the target forum.
Personally upgrading the forum is a much more sensible and future proof way of enabling mobile browsing vs. using third party apps but it has a following.