Hello David!
An edge router is a router that is found at the “edge” of a corporate or an enterprise network, that is, at the location where you connect to the “outside world” be it the Internet or a private or public WAN. Edge routers are usually designed to handle large amounts of traffic for hundreds or even thousands of users. Additionally, edge routers have many functionalities that would be useful at the edge of a large network such as security, content filtering, dual or multi homed connections (connecting to multiple ISPs for redundancy) and others.
Branch routers on the other hand are smaller, cheaper and are used primarily for remote sites (or a corporate branch office, which is where it gets its name from). They can also be used for a standalone office comprised of a small number of end users. The number of users branch routers can accommodate is usually several dozen and won’t often surpass 100. These routers usually have capabilities such as VPNs to connect to the corporate network and some security features. In general, they are a lower end devices as compared to network edge routers.
There really isn’t a solid line that separates the two types of routers. It all depends on the needs you have at each site. These needs however, more often than not, are derived from the number of end users you have at the site you want to provide for.
I hope this has been helpful!
Laz