To board a dinghy, a pirate simply walks to it, then presses Shift. They will then be offered a menu of ships they can take the dinghy to.
The menu will display the available vessels, their hull and sail health and their captain (should the vessel not belong to the pirate themselves).
If the pirate is in a crew or a guild member has a ship at sea, the pirate can then go directly to that vessel if allowed. Also, if another pirate has set their boarding level to Public, any pirate can board that vessel.