Discard is not a keyword ability like Trample or Flying, however it is a keyword in that it is defined in the comprehensive rules. In Magic it means to take a card in your hand put it in your graveyard. (Discarding means similar things in other games.)

Unless specified the person that is discarding a card gets to choose which one to get rid of. There are obviously plenty of effects where a player other than the one doing the discarding gets to choose. Distress is an example where what is discarded is not up to the player losing the card.

Discarding is not always the application of an effect, it is also often the cost associated with playing a spell or ability. In order to pay a cost, you must be able to discard the correct number of cards. You cannot activate Zombie Infestation‘s ability if you only have one card in hand.

To discard your hand means to put all of the cards in your hand into a graveyard. If this is a cost to activate an ability you can discard a hand of zero cards. For example, you can activate Slate of Ancestry‘s ability if you do not have any cards in hand.