Cost

There are two key concepts in the Yu-Gi-Oh! TCG concerning Costs: 1. Costs are paid before the activation of a card, not after. 2. You never get a "refund" on a cost. Costs are paid before activation. So to activate "Seven Tools of the Bandit", the effect of "Cannon Soldier", or play "Premature Burial", you pay the Life Points or Tribute the monster before you activate the card or use the effect. (In other words, you pay the cost then announce your intention to use the card.) This includes flipping the card, for a Trap, or playing it from your hand, for a Spell Card. If a card is negated (or, for a Continuous card, if it is destroyed in a chain), it makes no difference regarding the cost, because the cost has been paid. You do not receive a "refund" simply because your card was negated, or its effect was negated, or it was destroyed (in the case of a Continuous card).

Also, cards like "Dark World Dealings" present no cost, because the part that says you must discard is after the effect of drawing a card from your deck. This is one way to determine whether a card requires a cost or not. Here are some specific examples:

Example #1: Negation of "Seven Tools of the Bandit"

 * Step 1: Player A activates "Mirror Force".
 * Step 2: Player B pays 1000 Life Points, and activates "Seven Tools of the Bandit".
 * Step 3: Player A pays half his Life Points and activates "Solemn Judgment", negating "Seven Tools of the Bandit".


 * Step 3 resolves: "Solemn Judgment" negates and destroys "Seven Tools of the Bandit".
 * Step 2 doesn't resolve because the Trap Card was negated. There is no refund of Life Points. The Life Points were paid already. You don't get them back.
 * Step 1 resolves: "Mirror Force" resolves its effect.

Example #2: Negation in a Chain of "Premature Burial"

 * Step 1: Player A pays 800 Life Points and activates "Premature Burial". Player A chooses the "Jinzo" in his Graveyard as the target of "Premature Burial".
 * Step 2: Player B discards 1 card from his hand and chains "Magic Jammer" to negate "Premature Burial".
 * Step 2 resolves: "Magic Jammer" negates "Premature Burial".
 * Step 1 does not resolve because "Premature Burial" was negated. No monster is Special Summoned. There is no refund of Life Points because they have already been paid.

Example #3: Destruction in a Chain of "Premature Burial"

 * Step 1: Player A pays 800 Life Points and activates "Premature Burial". Player A chooses the "Jinzo" in his Graveyard as the target of "Premature Burial".
 * Step 2: Player B activates "Mystical Space Typhoon" to destroy "Premature Burial"
 * Step 2 resolves: "Mystical Space Typhoon" destroys "Premature Burial".
 * Step 1 does not resolve because "Premature Burial" was destroyed, and, as an Equip Spell Card, relies on the card continuing to be in play in order to achieve its effect. No monster is revived. There is no refund of Life Points because they have already been paid.

Example #4: Chaining "Mask of Restrict" to "Cannon Soldier's" effect

 * Step 1: Player A Tributes a monster and activates his "Cannon Soldier's" effect during his Main Phase 2.
 * Step 2: Player B activates "Mask of Restrict" as a chain to "Cannon Soldier's" effect.
 * Step 2 resolves: "Mask of Restrict" is in play and prevents Tributing of monsters.
 * Step 1 resolves: The monster was already Tributed before "Mask of Restrict" was successfully resolved. Therefore "Cannon Soldier's" effect resolves and Player B loses 500 Life Points. However, now that "Mask of Restrict" is active, further Tributes cannot take place.

More Info for new players
Costs don't always have to be in Life Points. Monsters can be Tributed, cards can be discarded from the hand, etc.

Costs as a Discard
Cards like "Dark Core" and "Lightning Vortex" have a discard cost of 1 card from your hand.

Costs as a Summoning Requirement
Cards like "Chaos Sorcerer" require you to remove 1 LIGHT and 1 DARK monster from play from your Graveyard.

You cannot use a card like "Disappear" to remove one of those monsters from play before it's removed for the Summoning, or a card to remove a card from your opponent's hand before he/she does, because you cannot beat a player to paying a cost.

Other Costs
Other costs include: discarding from the Deck or Tributing a monster(s). Sometimes, though it is not stated, the positive effect of a card can be a cost in itself, such as with "Astral Barrier".

Likewise, if a player activates a card with a cost affixed to it, and another player negates and/or destroys it (with "Magic Jammer", "Seven Tools of the Bandit", etc.), the first player must still pay the cost, even though the effect was negated.

You cannot partially pay a cost. For example, if you're forced to discard 2 cards, you must discard 2 cards, unless another card effect bypasses this, or it is otherwise impossible to do so, such as when you have one or less cards in your hand. In the latter case, you would not be able to pay the cost, since it has to be paid in full upon activation. A cost is similar to a Compulsory effect. There are very few cards that can negate costs (for example, "Spell Economics").

Misconceptions
Some cards can be "deceptive," cards can be worded to make someone, believe they are paying a cost when in reality they are not.

Here are some specific examples:

Example #1: Negation of "Two-Pronged Attack"

 * Step 1: Player A activates "Two-Pronged Attack".
 * Step 2: Player B negates "Two-Pronged Attack" with "Seven Tools of the Bandit"

Because destroying 2 of your own monsters is part of the entire effect not separated by a comma or a period, it is not a cost and your monsters are not destroyed.

Example #2: Negation of "Future Fusion"

 * Step 1: Player A actives "Future Fusion".
 * Step 2: Player B negates "Future Fusion" with "Dark Bribe"

In this card's case, although it is separated by 2 commas, there is no indication or suggestion that sending cards from your Deck to your Graveyard is part of an activation cost or requirement for this card.