Well, hopefully I won't contribute to 6 pages, but a couple things I wanted to note....
Breaker may be a bad "first example" of using Priority. 
Exiled Force <cringes /> would be a better one.
You summon 
Exiled Force. Your opponent is chomping at the bit with their 
Trap Hole waiting to make you lose it, but as the turn player. You have the priority to repond first to the summoning of Exiled. In this case you would use your priority to activate Exiled's Effect, tributing him to destroy a monster on the field, and thus leave your opponent high and dry with their 
Trap Hole.
The "best" way to do this in a duel is to say "I summon 
Exiled Force and use my priority to activate his effect. That way your opponent knows not to reveal any hidden 
Trap Holes or 
Torrential Tributes they have on hand. Granted it would be tempting to watch them flip it and then say "No no, I'm going to use my priority." I personally find such tactics highly disrespectful.
Similarly, if your opponent summons 
Exiled Force and does nothing, I would simply ask "Are you going to activate Exiled's effect?" If they respond with a No, then at that point you can effectively say, "Okay, I will repond to your summon with 
Trap Hole." I know there are MANY sticking points on this, because your opponent will most likely back peddle, but if you ask me, they have effectively 
Relinquished their priority at that point. Additionally, you can't chain Exiled's effect to the 
Trap Hole, because 
Trap Hole's effect is spell speed 2, and you can't chain a slower effect to a faster one.
Moving on, Breaker is an example of how the turn player "loses" their priority. When Breaker is summoned, he has an effect that triggers due to the summoning and activates immediately. Because this effect activates immediately and is out of the turn player's control, the opponent now has the opportunity to chain to Breaker's effect. Most likely they will chain with a 
Bottomless Trap Hole. Since BTH is a spell speed 2 trap, you are now out of luck to respond with any effects.
This is also shown in the 
Guardian Sphinx example initially given.
I hope this helps a bit. i'll be quiet now 8^D