What are the old time boats called with a sail?

It depends on their size and rig, and a "boat" is a small vessel, often oar-propelled, used as a ship's lighter or by fishermen and ferrymen!

The largest square-rigged merchant sailing-ships were barques and clippers; the latter built for speed as well as cargo capacity. (Strictly speaking, "ship" was itself originally a class of vessel by rig. )

A Schooner is a smaller, three-masted vessel with a mixture of triangular or quadrilateral and square (actually rectangular) sails.

Plus a few other, smaller ones such as ketch and yawl (2 masts, similar sail arrangements).

The Royal Navy developed different names for their ships, in the 18C. The larger ones were classified by "Rate" based on the number of guns carried.