I feel like clearing stuff up since apparently, people like to give bad examples and misinformation...:
1: Was somewhat explained already. Ignition effects are optional effects that the turn player can choose to activate in either one of his Main Phases, and only in the Main Phases. It's Spell Speed 1, meaning it can't be manually chained to anything.
(Someone mentioned Stardust Dragon here, but Stardust's effect to negate something that destroys, isn't an Ignition effect, so bad example)
2: Continuous effects are effects that are applied immediatly after a monster is succesfully summoned, such as Jinzo or Bountiful Artemis (I say succesfully summoned, because your opponent can use a trap card, such as Solemn Warning, on Jinzo even though his effect prevents trap cards from being activate, because Jinzo wasn't succesfully summoned). As Train Hardnett said, Continuous effects do not activate, they are just applied when they hit the field.
(Here, someone mentioned Monarchs as an example, but Monarchs do not have Continuous effects; they have Trigger effects)
3: Priority is not something you "call". It's something a player has.
Usually, turn player has priority, but it is passed if turn player starts a chain (actions that don't start a chain such as drawing during the Draw Phase, setting a card, etc., do not pass priority) or if the turn player states he is passing priority.
During the main phase, when a chain resolves and no monster is summoned, the turn player retains priority to activate a Spell Speed 2 or higher effect in response to the resolution of that chain. If the chain results in a monster being summoned, the turn player retains priority to activate an Ignition effect or Spell Speed 2 or higher effect in response.
Example 1: The turn player activates Call of the Haunted targeting Lonefire Blossom. When CotH resolves, Lonefire is special summoned to the field. The turn player can now retain priority to activate Lonefire Blossom's effect.
Example 2: The turn player activates Destiny Draw discarding Destiny Hero Malicious. When Destiny Draw resolves, the turn player has priority to activate a Spell Speed 2 or higher effect. They cannot retain priority to activate Malicious' effect. Here, the non-turn player would be able to chain D.D. Crow either to Destiny Draw, or activate it in response to the resolution of Destiny Draw, so that he can remove Malicious before its effect can be activated.
In regards to priority when a monster is summoned: If the turn player summons a monster, and there are no Trigger effects interfering, the turn player retains priority to activate the Ignition effect of one of their monsters, or activate a Spell Speed 2 or higher effect in response to the successful summon.
Example 1: The turn player summons a Snipe Hunter. The non-turn player has a lot of face down cards he would like to activate, however, the non-turn player must wait until the turn player decides whether or not he will activate the Ignition effect of any of his monsters, or a spell speed 2 or higher effect in response to the summon. After the turn player has performed one of those actions, or has passed priority to the non-turn player by deciding that they will not activate any effects, the non-turn player is allowed to activate a spell speed 2 or higher effect in response to the summon of Snipe Hunter.
Example 2: The turn player successfully summons an Exiled Force while they control a Cyber Valley. The turn player can use his priority to activate either one of those monsters' Ignition effects.
(In the Lonefire vs. Torrential example you gave, Placido, the summoned monster will not be destroyed. Amy, in your example, 2 things are wrong: 1st, if you activate Black Rose's effect to nuke the field when it's summoned, you don't chain it to the summon, because synchro summoning a monster doesn't start a chain. 2nd, you state that it's a Trigger effect, yet you're saying you're using your priority to place it as Chain Link 1 (which btw, wouldn't be possible if the summon started a chain), but as I've stated above, when a monster is summoned, you retain priority to activate Ignition effects, not Trigger effects, so whether or not you want to activate its effect to blow up the field, has nothing to do with priority)