String is generally single-stranded, and is easily frayed. This works well for projects that will have a fringe, such as earrings or some wall hangings that you will “brush out”.
Rope is generally used to refer to multiple strands twisted together. You will see this referred to as “3 ply twisted rope” or “4 strand twisted cotton”. This is more suitable to projects that need more strength (like a plant hanger). Another aspect of twisted rope is the wavy-like fringe it has when unwound.
Cord is generally woven - you will see “braided cord” on many of the listings. One thing to keep in mind when using this type of macrame cord is that it may not be cotton - some braided cord is propylene (which is more suitable to outdoor projects). This cord is also much more difficult to unwind for creating a “fringe”.