You can trade items to other players or, if for a particular quest, to NPCs. Select player/NPC that you wish to trade with and select the trade option in the menu bar. Move the item(s) or gil that you wish to trade into the trade boxes and confirm the trade.
When trading with another player the trade will not be confirmed until both players confirm the trade. When trading to an NPC, the NPC will simply take the item - if he wants it.