The glue we use in the video is a urea-resin-type glue. It's a powder which is mixed with water, and which is water-clean-up until it hardens. While it no longer is called waterproof (it used to be called this, but the standard has changed), it's far more waterproof than we need for our boats. It doesn't pass the boiled-joint test; don't put your boat in boiling water if you use this glue. This kind of glue is also called Plastic Resin Glue. You can find it at ACE Hardware stores; they can order tubs of their house brand glue if it's not in stock. You can find the Weldwood Plastic Resin glue online at Aircraft Spruce & Specialty.
An easier type of glue to find and to use is an aliphatic resin glue like Titebond III. These "yellow glues" are simple one-part, water-clean-up adhesives which make great wood joints. You can use this type of glue for boats which are stored out of the water most of the time.
We think the BEST choice is epoxy resin. This also works great, and can be used to do the fiberglassing also. As with the urethane glue, epoxy can be allergenic so keep an eye out for this reaction. There's a school of thought which claims that the only type of glue to use for boatbuilding is epoxy. As with any extremist belief-system, it's not true. Epoxy is great, but any of these three glue types will work fine—use what you like!