How does the priority rule work in the newly relased Yu-Gi-Oh app?

Well, priority is different then what it was back in the TCG/OCG. As of now in the TCG, priority applies to scenarios only like when the turn player starts Main Phase 1 and summons a monster. If the opponent chooses not to respond, then turn player has the option to freely do his next move without the opponent randomly activating a trap or quick effect. That's the priority the turn player is given. The opponent would have to wait for a chain or response to anything else he does.
In Duel Links, it's quite similar. Picture the same scenario and opponent chooses not to respond to the turn player's summon. You're not able to activate any quick effect, spell speed 2 cards, or traps until the turn players either does something else or switches phases. That's the priority a turn players has now. No need to learn how it was back then, as it may confuse you if this hasn't yet lol :P

What exactly do you mean by priority?