Problem Solving Card Text

PSCT stands for Problem Solving Card Text and it is the new type of text Konami uses for writing cards. It makes the game a whole lot easier and solves a lot of rulings disputes (Older cards will not necessarily follow these rules.).
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It is easy to tell which card effects stars a chain link (and, of course, as always, spells and traps start a chain).

Sangan:
When this card is sent from the field to the Graveyard: Add 1 monster with 1500 or less ATK from your Deck to your hand.
Gemeni Spark:
Tribute 1 face-up Level 4 Gemini monster you control to target 1 card on the field; destroy it and draw 1 card.

When a card is activated, it may have a cost, or you have to choose a target for it, and both of these actions are done at the same time as activating the card.
If an action is written before a semi-colon, it is done at activation of the effect. If it is written after a semi-colon, or there is no semi-colon, it is done when the effect resolves.
So basically, the thing before the : is the timing (If we make a chain, do I do things when I activate my card or when the card resolves?)
The thing before the ; is the targeting and the costs.
Anything after the : or the ; (the last thing) is what actually happens (resolution).

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There are lots of cards with effects that negate special summons. However, these can only negate special summons which do not start a chain. To prevent the Special Summon effect that starts a chain, the effect itself must be negated.

Examples of cards that negate Special Summons: Thunder king and Laggia

Special summons:
Chaos Sorcerer:
Cannot be Normal Summoned or Set. Must first be Special Summoned (from your hand) by banishing 1 LIGHT and 1 DARK monster from your Graveyard. Once per turn: You can target 1 face-up monster on the field: banish that target. This card cannot attack during the turn you activate this effect.
We see that its Special Summon effect does not start a Chain (no colon or semi-colon) so cards like laggia could negate its Special Summon!

Gorz, Emissary of Darkness:
When you take damage from a card your opponent controls: You can Special Summon this card from your hand. You must control no cards to activate and to resolve this effect. When Special Summoned this way, activate the appropriate effect, based on the type of damage.
● Battle Damage: Special Summon 1 "Emissary of Darkness Token" (Fairy-Type/LIGHT/Level 7/ATK ?/DEF ?). Its ATK and DEF are each equal to the amount of Battle Damage you took.
● Effect Damage: Inflict damage to your opponent equal to the amount of damage you took.
The special summon does not start a chain. In order to prevent the summon, you must negate the gorz's activation of effect. Something like Solemn Judgment could not do so because it cannot negate monster effects.

Call of the haunted:
Target 1 monster in your Graveyard; Special Summon that target in face-up Attack Position. When this card leaves the field, destroy that target. When that target is destroyed, destroy this card.
We see that it starts a chain, and so to prevent the special summon a card would have to negate the activation of call itself. Something like Thunder King Rai-Oh could not do so because it cannot negate trap cards.
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If they use the word targets after the semi-colon, any conditions on the chosen cards must still be there when the effect resolves, and if it doesn't use targets then the cards chosen only need to be in the same place.
Additionally, if it uses the word both, then if one chosen card is no longer correct no others are affected. Otherwise, if one chosen card is no longer correct, the others are still affected.
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Conjunctions:
Then (Do A, then do B)- B happens after A, even though they’re part of one card effect. These things happen in sequence, not simultaneously. A is required for B, but not the other way. If A does not happen, then stop. If B cannot happen, you still do A.

Also (Do A, also do B)- considered simultaneous. Both happen at the same time. Neither is required for the other.

And if you do (Do A, and if you do, do B)- A is required for B. A does not happen, then stop. If B cannot happen, you still do A.

And (Do A and B)- Considered simultaneous. Both happen at the same time.BOTH are required. If you cannot do both, then you do nothing.

There's multiple conjunctives sometimes but that will be a different lesson. I hope this helped you understand cards more.

Thank you to Konami for making PSCT!