The complete rules for priority, from the MTG Comprehensive Rules, are as follows:
117.3. Which player has priority is determined by the following rules:
117.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3).
117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
117.3c If a player has priority when they cast a spell, activate an ability, or take a special action, that player receives priority afterward.
117.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, they announce what mana is there. Then the next player in turn order receives priority.
117.4. If all players pass in succession (that is, if all players pass without taking any actions in between passing), the spell or ability on top of the stack resolves or, if the stack is empty, the phase or step ends.
117.5. Each time a player would get priority, the game first performs all applicable state-based actions as a single event (see rule 704, “State-Based Actions”), then repeats this process until no state-based actions are performed. Then triggered abilities are put on the stack (see rule 603, “Handling Triggered Abilities”). These steps repeat in order until no further state-based actions are performed and no abilities trigger. Then the player who would have received priority does so.
117.6. In a multiplayer game using the shared team turns option, teams rather than individual players have priority. See rule 805, “Shared Team Turns Option.”
117.7. If a player with priority casts a spell or activates an activated ability while another spell or ability is already on the stack, the new spell or ability has been cast or activated “in response to” the earlier spell or ability. The new spell or ability will resolve first. See rule 608, “Resolving Spells and Abilities.”
Note that the rules don't describe this as a "round" of priority—there are no "priority rounds". Rather, the active player starts each step and phase of their turn with priority, and each other player is given priority, in turn, in order to respond to plays made by the other players. Effects resolve, and steps and phases end, only once all players have passed priority.
To make this more concrete:
It is Player A's turn. Player A has priority and casts Deadly Dispute. Deadly Dispute has the text "As an additional cost to cast this spell, sacrifice an artifact or creature." Player A sacrifices some token creature and puts Deadly Dispute on the stack.
Player B has a Blood Artist on board, which has a triggered ability that cares about creatures dying, so this ability goes on the stack. Player C, for some reason, has a Mold Adder on the board, which has a triggered ability that cares about opponents casting spells, so that ability goes on the stack.
At this point, Player A still has priority.
Player A passes priority to Player B, and state based actions occur.
Player B passes priority to Player C, and state based actions occur.
Player C notices that Player A just sacrificed their token, and now the only creature they have on board is their commander, Kambal, Counsul of Allocation. Player C decides to cast Diabolic Edict, targeting Player A, in order to remove their commander. Kambal has a triggered ability which cares about opponents casting spells, so that goes on the stack.
Player C passes priority to Player A, and state based actions occur.
Player A doesn't want to lose their commander so, noticing that Player C is now tapped out, they cast Mana Tithe (so that goes on the stack). The Mold Adder trigger goes on the stack, and Player A passes priority to Player B.
Player B doesn't have anything to do, so they pass priority to Player C.
Player C has no further actions, so they pass priority, and the stack starts to resolve. The Mold Adder trigger resolves first, and Player A gets priority.
Assuming that no one has any other actions, each ability on the stack resolves, and Player A ends up with priority.
Again, there are no "rounds" of priority. Priority passes from player to player until each player passes without doing anything. Every player gets an opportunity to react to something done by another player, but they don't get a chance to react to their own abilities or spells after they pass priority, and everyone else at the table passes priority, too.
