Flex duct work is used in many cases for HVAC installations. The tubing, as its name indicates, is flexible, which allows it to bend and weave between wood beams and other obstacles in a crawl space or attic area. The tubing of flex duct work is plastic with a layer of insulation wrapped around it. The entire tubing is then sealed inside a vinyl vapor wrap to keep out moisture.
Metal ducts are constructed from sheet metal and are not flexible in any way. They are built in a rectangle shape and attached together with sheet metal screws. In some cases they are also wrapped in insulation to improve efficiency. Although as a whole this duct will stay together well, it is more likely to leak due to its many connection points. Metal ducts are normally made from steel or aluminum, but sometimes they come in copper if the ducts are going to be exposed to salty air.
Duct board is made of fiberglass. It is a lightweight installation, but fairly expensive. The board itself can be damaged easily by a careless worker, but once in place it lasts a long time. Leaks are less likely than with sheet metal, but still more likely than with flex duct.
In times past, you might have seen ducts made with asbestos. This has since been proven to be hazardous and is no longer used. If you have an older home, it might be a good idea to have this checked out to ensure your family's safety.