Cost

Costs (either summoning or activation costs) never go to the stack and must be paid immediately (upon summoning or activation).

The activation cost of an effect is mandatory, and if you cannot pay it, you cannot activate the effect. Summoning costs must be paid if possible, and if not, the summon is fizzled. If an entire summoning cost cannot be paid, you do not need to pay partially. You must declare how you are paying all parts of a cost before resolving any triggers associated with the payment. Any triggers caused by paying a cost go to a proc'ed queue to be declared into the stack once the declaration or summon is finished.

Typically, costs fall into these categories:

Mana Cost

Requires the removal of mana counters from cards you control. If the effect says "consume {X} mana", that means it must be paid with mana collected in this card. If it says "consume any {X} mana", you can use mana from any cards you control. In that case, you can remove any amount of mana of each card as long as the total is equal to the cost.

You can also sac face-down cards to pay "any" mana costs if the card has a non-zero Will, its Will counting towards the cost (see Hidden Sacrifice). Cards sac'ed this way are never summoned. Using this mechanic, you can overpay but only at the last payment parcel.

Sacrificial Cost

This requires you to "sac" (sacrifice) cards. Sacrifice means destroy a card you control. It counts as normal destruction for all intents and purposes. It can be a face-up or face-down card, as long as it is currently on the field and matches the requirements (if the requirements specify any aspect other than card type, they must be face-up as face-down cards do not have any other properties).

Note that cards that are in the process of being summoned (but have not have its summoning costs paid yet) are not considered to be on the field and cannot be sac'ed.

Discard

Discard means "send a card from your hand to the graveyard".