What are the ropes that support and control the sails of a ship called?

Halyards are the lines (ropes) that raise sails. Downhauls lower them (note that they are not always included as gravity does usually help out, though they are sometimes necessary).

Sheets are used to pull the bottom corner of a sail aft (or towards the "stern" or back of the ship). Tacks pull them forward (ie towards the "bow" or front of the ship).

Square sails also have lines that are used to douse (square sails are often "lowered" by pulling them up, and set by letting them fall down as they are attached on the top edge by a long horizontal spar called a yard) or to reef them (decreasing the sail area in cases of heavier wind) These are the buntlines, clewlines, reeflines, and reefing points.

The yards that hold square sails up also need to be able to turn, and lines attached at their ends called braces pull either the port or starboard (left or right) end of the yard aft.

There are, of course, many more however these are probably the most important.