I'm from New England and schooners (along with other aspects of fishing/sailing/maritime history) featured prominently in discussions of local history. But why was the schooner specifically so popular and successful in this region as opposed to other types of ships?
There are a couple reasons:
Labor Shortages. For much of the early history of New England labor shortages were a real and common issue. After the Great Migration of the 1630s we begin seeing large numbers of English colonists heading back to England (this is one reason slavery takes off during this period). For a larger square rigged ship, such as East Indiamen, you would need a crew of 10+ to man shifts during calm seas; a topsail schooner can be manned by a crew of 3, or even one in favorable weather or an emergency.
Logistics of building. Within the first few decades of settlement in New England timber forests near the coast were already being exhausted. Building shallow drafted vessels further inland allowed the cost of transporting timber to remain lower, and allowed these yards to remain closer to the mills. All of this helped keep cost even lower.
Cost. Besides the obvious savings on crew, schooners can generally be built quickly, make more efficient use of sail compared to a ketch or barque, and are relatively simple to perform maintenance on. All of this made them extremely popular for fisherman and sailors who were trying to move up financially; it was quite common for a group of sailors/fisherman to become part owners with family members or friends.
They were practical Schooners are built with a shallow draft, meaning they can access shallower waters. This gave greater access to inland waterways, meaning goods could get farther inland without having to be crossloaded to smaller boats or wagons - keeping the cost of transport and thus goods lower. Despite this, they also handled fairly well in blue water; allowing them to take part in the highly profitable West Indies trade.
Speed. The way schooners of New England are generally rigged means that more sail is being used at different wind angles than other traditional ships. They can sail closer to the wind than other vessels their size, and in general maintain greater speed during transit. Being able to travel at a good speed for both legs of a journey meant faster exchange of goods, which again meant greater profit to the owner: goods are on market faster, crews are paid for less time at sea, and market fluctuations have less of an impact on profit.
All of these reasons helped the schooner to not only dominate the coastal and West Indies trade into the 19th century, but also made them a near perfect vessel for privateers looking to prey on British merchants during the American Revolution and War of 1812.
If you want a little more reading on schooners, I'll suggest reading Dana Story's Shipbuilders of Essex, which documents the methods, families, and industry of Essex, Massachusetts; a place which dominated the North American schooner building industry for centuries. There is also a 1947 documentary of the same title that is floating around on youtube that is worth a look.
If you prefer a more hands on approach, the Essex Shipbuilding Museum is a wealth of knowledge and houses the last traditional wooden shipyard in the United States, where NEA Fellow Harold Burnham still builds in the old way. Harold is a fantastic resource and a hell of a nice guy, and you can also sail on a couple of schooners he built; the replica privateer Fame out of Salem, as well as the Thomas E. Lannon and Ardelle(his own person schooner) which sail out of Maritime Gloucester.