For someone who works with machines that are for large, heavy-duty industrial processes (like steel mills), the answer will be different than if they come from a background of moderate process requirements (pumps and fans) or precision processes (servo motors, actuators, etc).
From my perspective (which is the "large industrial" side of things): I work with designs that have continuous torque ratings that start at about 4500 lb.ft (6.1 x 10^3 N.m) and go up from there. The largest continuous torque rating I've ever built is rated at 5252000 lb.ft (7.1 x 10^6 N.m). That big one had a PEAK torque rating of 3.5 per unit ... or roughly 25.0 x10^6 N.m!
A standard squirrel cage motor has 60% starting torque and 175% maximum (or breakdown) torque. A high torque motor can have 100 to 150% or more starting torque and from 200 to 300% or more maximum torque. It depends on the application. The motor manufacturer can design the motor for higher torque to meet the requirements of the load. There is a definition on motor category related to torque on NEMA MG1.
As above mentioned, it is possible to rank machines based on transient torque performance as well as continuous torque. Transient - in this case - refers to the acceleration / deceleration process (starting or "locked rotor" torque, pull-in torque, breakdown or pull out torque, etc.). These are often given as a percentage of rated continuous torque. They can range from a fraction of rated torque (20-30 percent) up to several times the continuous rating (175 - 350 percent). If this is where you're looking, then a "high" torque device is something that offers significantly more than the "normal" rating does.
Now I said I worked with some pretty large machines. If I worked with servo motors and actuators, I might consider a "high" torque design to be something rated continuously at 2 lb.ft (2.7 N.m) and an "ultra high" design to be rated around 15 lb.ft (20 N.m). Peak torque requirements might range from 50 percent up through 200 percent under specific transient conditions.
In terms of industrial applications, a "torque" motor is often one that rotates at a VERY slow speed - most times, less than 1 rpm and often it never makes even 1 full revolution! Another way to look at it is that the motor operates at what is effectively a "continuous stall" condition. This requires some extra design consideration to maintain adequate cooling of the all the machine components, as the rotor is doing absolutely nothing to help move the primary coolant (air) around inside the enclosure.