What holds the mast
The mast sits in the mast step. The forestay runs from the masthead to the bow. Some Megin dinghies also have shrouds to support the mast. The fixed parts of the rig are called standing rigging.
The parts you handle
Halyards are used to hoist the sails and sheets to control them. The sprit holds out the upper aft corner of the mainsail. These parts belong to the running rigging.
An overview of the equipment
Brejning Efterskole's handbook contains an overview of the school's dinghy equipment, a checklist and pictures of the rudder, mast and sprit, among other things. It describes the school's particular equipment and lending arrangements. See the original for the pictures and full explanations.
Finn R. Andersen's handbook goes into more detail about the parts and their names.