Chains are Complicated.

You just activated Fire Formation - Tenki. You're opponent activated Mystical Space Typhoon, in chain. You chain Magic Drain. You're opponent finishes the chain with Dark Bribe.

You're, probably, totally confused at this point. If not, you're about to be.

The first thing that I'm going to go over is spell speeds. Every single card in the game has a Separate Spell Speed. A Spell Speed tells you when you can activate this effect, how you can respond to that effect, and how you can respond to other effects using it.

The first speed I'm going to go over are Spell Speed 1. Spell Speed 1 card effects are the slowest of all effects, and are very rarely higher than Chain Link 1. The only way this is Chain Link 2 or higher, is if something happens to trigger 2 of them at the same time. Cards in this tier of Spell Speeds cannot be chained in response to the activation, but can activate simultaneously.
The following cards Spell Speed 1:
Normal Spell Cards
Field Spell Cards
Continuous Spell Cards
Ritual Spell Cards
Equip Spell Cards
Monster Flip Effects
Ignition Effects
Trigger Effects
Ignition-Like Effects
Trigger-Like Effects

Normal Spells, Continuous Spells, Equip Spells, Ignition-Like Effects and Trigger-Like effects can only be activated during the Main Phase of the Turn Player's turn, during an Open-game state. This means that the only thing that has to be going on in order for you to activate one of these is absolutely nothing. That's right. Nothing needs to be going on in order to use these effects.
If multiple instances of a Trigger effects and Trigger-like effects are triggered at the same time, they would go off simultaneously. If they were both the Turn Player's card effects, the turn player would be able to decide in which order the chain went off in.
So, lets look at this situation. T.G. Hyper Librarian's Effect reads as followed, and is Spell Speed 1.
1 Tuner + 1 or more Non-Tuner monsters
When a monster is Synchro Summoned: Draw 1 card. This card must be face-up on the field to activate and resolve this effect.
So, that means that, during either players turn, during the Main Phase, if a monster is Synchro summoned, you get a card.
Now, let's look at Formula Synchron; another ignition effect.
1 Tuner + 1 or more Non-Tuner monsters
When this card is Synchro Summoned: You can draw 1 card. During your opponent's Main Phase: You can Synchro Summon using this face-up card you control as a Synchro Material Monster.
So, that means that, when this little guy is Synchro Summoned, you get a new card in your hand. The other effect isn't important right now, so we'll ignore it.
If you Synchro Summon Formula Synchron with T.G. on the field, you would normally be crying because both cards say when. That means that they miss timing, which I will go further into later on. But, because they are both Spell Speed 1 effects, they activate simultaneously.

Triger-Like Spell/traps and Trigger Monster both activate at any time that their requirements for game state is met. Example: Skilled Dark Magician

Ignition-like Spell/traps and Ignition Monster effects are effects that can be activated at any time during the appropriate phase, and are normally accessible after a certain condition has been met. Example: Call of the Mummy

Monster Flip effects are activated during the Damage Step, before the monster is destroyed. They can also activate during any time in the turn that they are flipped from face-down to face-up.

Please note: Spell Speed 1 cards can have Spell Speed 2 effects. For example, a Continuous Spell card may have an effect that can be used during your opponent's turn.

The next type of Spell Speeds are Spell Speed 2. These Spells can be activated during either players turn, and can chain to Spell Speed 1 cards and Spell Speed 2, but cannot be chained to by Spell Speed 1 cards. They are often referred to as 'Fast Effects'.

The following cards are Spell Speed 2:
Quick-Play Spell Cards
Continuous Trap Cards
Normal Trap Cards
Quick Effects
Quick-like Effects

Quick-Play Spell Cards can be activated from the owner's hand during the players turn, at any time during the turn. Quick-Plays cannot, however, be activated from the hand during the opponent's turn. They must first be placed face-down on the field, in which they can now be played during your opponents turn. They cannot, however, be activated during the turn you put them on the field. If you set a Mystical Space Typhoon face-down, it is now acting like a trap-card; it can only be useful to you at the start of your opponent's next turn. Example: Mystical Space Typhoon.

Continuous Trap Cards can be activated from the owner's field during either player's turn, so long as the card has been face-down on the field for at least 1 turn. Continuous Trap cards have Ignition and Trigger effects that are referred to as Trigger-like effects and Ignition-like effects. This means that the card has effects similar to the monster counterparts, but are capable of using them during either players turn. That's right; these cards can use their devastating effect during your turn and your opponent's. Continuous Trap Cards are usable by their owners, and can effect both players, so long as it's owner can sustain it's presence on the field. Once it's removed, or the owner can no longer support it's cost, it is no longer accessible. Example: Dark Smog.

Normal Trap Cards can be activated from the owner's field, or wherever else they may activate, during either player's turn, so long as the card has been face-down on the field for at least 1 turn. Normal Trap Cards have Ignition-like and trigger-like effects that can be used upon activation, in which they are then sent to the graveyard (unless stated otherwise). They generally have a single use before going away, and they can be used during either player's turn. Example: Dimensional Prison

Quick Effects can be activated from the owner's field, or wherever else they may activate, during either player's turn, so long as the proper conditions have been met. Quick Effects are monster effects that can be used when the game state has changed to meet the requirements needed to activate. They generally require some kind of cost to be paid whenever their game state is made, and they then stop or add on too whatever just happened. They can be considered Counters in the form of monsters. Example: Stardust Dragon

Quick-like Effects are trap cards that can be activated from the owner's field during either player's turn, so long as it has been face-down for at least 1 turn. Quick-like effects have separate effects that can be activated, in addition to the activation of the card itself. So you you could start 2 chains with the exact same card, simply by activating the card and then activating it's effect. Like-wise, your opponent could chain to both of the effects. An example would be Dark Smog, which can be used to discard 1 card to banish a monster from your opponent's grave. This is a Quick-like effect that can be activated, in line with the fact that it's a trap card all in itself. Example: Skull Lair

The final speed is Spell Speed 3. Spell Speed 3 cards can chain to all other chains, including itself, but cannot be chained to by Spell Speed 2 or Spell Speed 1 cards. Spell Speed 3 cards consist of one type of card:
Counter Trap Card

Counter Trap Cards can be used during either player's turn, so long as the card has been face-down on the field for at least 1 turn. Once the Counter Trap Card has been activated, most chains end on the spot. Counter Trap cards are activated to completely stop the activation or use of the card you just tried playing or using. Being the fastest Spell Speeds out there, the only thing that you can use to stop a Spell Speed 3 card is another Spell Speed 3. Example: Solemn Warning

Sigh. . . All this, and then we have 1 Card to top them all. Super Polymerization. It cannot be chained to by other effects. . .


Anywho, now that you know about Chains, this chain doesn't look so darn frightening.

Tenki activates. Tenki is a Spell Speed 1 with another Spell Speed 1 effect, so it has two effects that are used upon activation. By activating the Spell Speed 2 MST, you destroy Tenki. You wanna stop this by activating the first counter Trap Card, Magic Drain. Your opponent then uses Dark Bribe to negate your counter trap card. Here, we have seen all 3 Spell Speeds in 1 chain.

But. . I still get my Beast-Warrior to hand. So. . .what's it matter?

Well, that's just it, sir. You don't get your Beast-Warrior.

Chains resolve Backwards. In other words, Tenki is Chain Link 1. So, that means, that each card that activates after Tenki is one Chain Link higher than the card before it. So, the Chain link would like something like this:

[Chain Start] Tenki (1) >> MST (2) >> Magic Drain (3) >> Dark Bribe (4) [Chain End]
I know what you're thinking now: Tenki gets its effect off before MST resolves, so you get your Beast-Warrior. NOT. Chains resolve Backwards, which means the latest cards activated resolve first. Since the chain was built like this, it would resolve like this:
(Resolution begins) Dark Bribe (4) >> Magic Drain (3) >> MST (2) >> Tenki (1) (Resolution Ends)
Dark Bribe resolves and negates Magic Drain. Magic Drain then attempts to resolve, but it's effect is now null; it's resolution is useless. MST was not negated, because the card that attempted to resolve to stop it never resolved successfully. Tenki is destroyed.

But. . .Tenki said that whenever I activate it, which I did, I get a Beast-warrior
But. . .Tenki's other effect is a Spell-Speed 1 card that activates after it activates successfully. It would, normally, be able to go off, but it misses it's timing. The last thing that happened in the game state was not it's activation, but it's destruction.
Also, Tenki would also still have to be on the field in order for it's activation effect to work.

Now that that Chain is dealt with, let's talk about Missing Timing.

"If" Effects cannot miss timing. When a Card says "if" in it's effect text, that means that at any time during the game, when this condition is met, it will activate. You can't do anything to this card to make it's effect unusable. Your only hope is to negate it's effect. Otherwise, as soon as it's effect becomes accessible, your opponent can use it.
If Effects can activate at any time during a chain, and are not required to be specific with timing. So, they cannot miss timing. The only time they are not activated is if your opponent either forgets or chooses not too activate them.

"When" Effects, however, can miss timing. When a card says "When" in it's effect text, that means that whenever that condition is met, it must be the last thing in the game state to trigger the effect. In other words, if something happened after the when condition was met, the when effect cannot activate. The only instance that this is avoided is if a Card has a mandatory effect, which I'll get into later. One said example is E-Hero Absolute Zero.

There are cards that do not start chains. These cards are identified by a form of Problem Solving Card Text, or PSCT.

Monster effects, Spell effects and trap effects without : or ; do not start chains. These effects do not start chains, and therefor cannot be chained to. However, the Monster Summoning, Spell Activation and Trap Activation can be chained too. Even if their effects can't be chained to, the activation of the card itself opens a way for you too stop it.

Now, for Turn Priority.
Whenever a chain has begun, priority is the first thing that is looked into before anything continues. The first thing you need to know about Priority is this: During the Opponent's turn, they get Priority in each of their phases. That means you can't activate any effects or cards until your opponent has conducted their first action of each phase.
Don't quite understand? I didn't think you would, so I came prepared.
Say the game state is in the beginning of the Opponent's Battle Phase. Until their monster declares an attack, or the opponent activates a card/effect during the Battle Phase, you cannot activate any cards/effects. The same rule applies to each phase of the Opponent's turn; until they do something, you can't.

The second thing about Turn Priority is this: Whenever a Chain begins that involves two effects, both owned by the Turn Player, that would activate at the same time, the Turn Player gets to decide the order in which the Chain builds. For example, if an Opponent discarded an Atlantean Dragoons and MermailAbyss Gunde to special Summon MermailAbyss Megalo, then they would get to decide which effect activates first, and how the chain resolves. By the player's turn choice, they could either Special Summon by Gunde's Effect, Search a Sea-Serpent with Dragoons or Search a Spell/Trap card with Megalo, in any order they choose.

The third thing about Turn Priority is this: Whenever a Chain begins that involves two effects, both owned by separate players, that would activate at the same time, the Turn Player's effect will always activate first. That means that if a Monster effect on both sides of the field had it's condition met, and they could both activate there effects, the Turn Player's effect will be the lower chain link. The best example here is if both players use Honest. One attacks the other, and they both use Honest at the same step of the Damage Step, at the same time. Player A's Honest would be Chain Link 1, and Player B's would be CL 2. CL 2 would Resolve first, and Player A's monster would still win.

The final rule involving Priority is effect Priority. Certain effects activate before others do in their chain, and this is very important. At any time a chain begins, this would be how it went.

-Turn Player's Mandatory Effects
-Turn Player's Optional Effects
-Opponent's Mandatory Effects
-Opponent's Optional Effects

If a large chain was created containing effects from both sides of the field, this would be how it was resolved. Mandatory effects are effects that Say "If. . .,then" or "When. . .,then." These effects tell you that you don't get the option to do something; it has to happen. When effects normally miss timing terribly, but if they are Mandatory when effects, they have to activate, and cannot miss timing.
Optional Effects are effects that say "If. . .,you can" or "When. . .,you can." These effects are completely optional, and are not mandatory for you to activate. When optional effects can miss timing, and both types of effects will only activate if the owner wants them too.
As always, The turn Player has Priority over the Opponent, so their effects would activate before the Opponents.

Happy chaining.