Cost

A 'Cost' is usually a sum of Life Points required to activate, or maintain a card. Brain Control's 800 Life Points are a 'Cost', as are the 2000 Life Points needed to activate Final Countdown.

However, costs don't always have to be in Life Points. Cards like Dark Core and Lightning Vortex have a discard cost of 1 card from your hand. With Chaos Sorcerer, removing 2 monsters from play is a cost. You cannot use a card like Disappear to remove one of those monsters from play, before Chaos Sorcerer does, because you cannot beat a player to paying a cost. Other costs include: discarding from the Deck or offering Tribute. 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 your forced to discard 2 cards you must discard 2 cards unless another card effect bypasses this or it is otherwise impossible to do so. In the latter case of the example you would not have to discard. A cost is similar to a Compulsory effect. There are very few if any cards that can negate costs.