This mechanic usually requires players to pick up an item or good at one location on the playing board and bring it to another location on the playing board. Initial placement of the item can be either predetermined or random. The delivery of the good usually gives the player money to do more actions with. In most cases, there is a game rule or another mechanic that determines where the item needs to go.
Empire Builder is a classic pickup-and-deliver game from 1980 that remains popular today. In this game, players build railroads between cities, and move trains around on the track. Players hold contracts specifying that specific cities demand specific types of goods. To fulfill a contract, a player must travel to a city where that good is available, pick it up, and deliver it to the destination city. When the player completes the contract, the player receives money as specified on the contract.
For an overview and discussion of popular games with this mechanic, see: Pickup-and-Deliver Games: Your favourites and why
Pick up and deliver games fan