When instructed to discard a card, which cards can you choose from? Where do those cards go?

When instructed to discard a card, which cards can you choose from? Where do those cards go?

"Discard a card" without any other qualifiers means "put a Power Card from your hand into your discard pile." This most commonly occurs as part of paying for a Choice Event, but can also appear as a cost or detrimental effect.
Some effects (most notably, the standard effects of Time Passes) discard Power Cards from play because they interact with a specific card that's in play or with cards in play in general. In that case, put the Power Card(s) from play into your discard pile. You can't discard a Power Card from play unless otherwise instructed to do so.
If instructed to discard a card not controlled by a player, the card goes to that deck's discard pile. This most commonly occurs when gaining a Power Card, discarding the top card of the Event Deck on Turn 1, and when discarding a Minor Power as a randomizer.