Put the set aside card face up on the Gibbet to remember what it's for. You have to set aside a card; it's not optional. Gibbet and the card stay there until the start of your next turn, at which point you put the set aside card into the trash.
While the Gibbet is in play, before discarding cards on their Clean-up, each other player must trash a card from play. They choose which card, as long as it is valid. A valid card is one which has a higher cost than the set aside card, not an equal or lesser cost, or a cost which cannot be directly compared (as can happen with Potion costs). If the player does not have such a card in play, they do not have to trash a card.
Only their cards from play can be trashed. Cards which were kept in hand, or are set aside, are not eligible, even if they meet the cost requirement.
If you do not have a card in your hand to set aside, then play Gibbet alone. The requirement for other players to trash a card still stands, but they may pick any card without regard to cost.
If there are multiple Gibbets in play, a card must be trashed for each one of them.
It is possible to trash cards which are dependent on or modifying other cards which are not in play, thereby 'orphaning' them. The state of these orphan cards needs to be resolved, which is done by putting them into their owner's discard pile immediately.
For example, if a player trashes a Haven from play, the card set aside with it will go straight into their discard pile.
Cards on play mats, such as the Native Village mat, are unaffected.
The active player can choose what order they resolve effects which are triggered at the same time, so they can choose have the trashing occur after any other card's "Start of Clean-up" effects.
For example, at the start of their Clean-up a player may use the promotional card Walled Village to put itself on top of their deck, removing it from play and making it immune to the Gibbet (although it means another valid card might have to be trashed, instead).
A card which protects against attacks, like Moat, can be revealed when Gibbet is first played and will provide protection for as long as the Gibbet is in play. Revealing on later turns, once it is already in play, is too late. Use player tokens, as described in the Duration-Attack rules to remember that a particular player is immune to a Gibbet.
Gibbet is discarded in the Clean-up phase of your next turn.