Halyards are the lines that pull the sails up the mast. A typical small sloop like this sailboat has two sails, the mainsail and jib, and thus has two halyards – one to pull up the top corner ("head") of each sail. (We’ll see this is Part 2 of this course.)
At the end of a halyard is a fitting, called a shackle, that attaches the sail to the line. The line then runs up to a block (pulley) at the masthead, and comes back down alongside the mast as you see here. Pulling down on this end of the halyard hoists the sail up.
When the sail is up, the halyard is tied off tight to the mast cleat using a cleat hitch, as shown here.
Halyards are part of the boat’s running rigging. "Running rigging" refers to all the lines that control the sails or other rigging, which can be moved or adjusted while sailing - unlike the fixed rigging, the usually metal, fixed parts of the rig (mast, boom, stays, shrouds).