One newton-metre is exactly 100 newton-centimetres. It is a plain decimal shift within the metric system, and the smaller unit exists for the same reason the pound-inch does — small torques deserve legible numbers.
About the units
Both units express torque as a force times a distance from the axis. The newton-centimetre is one hundredth of a newton-metre, and it is used where the newton-metre would give values below one. Small electric motors, instrument bearings, medical devices and precision mechanisms all operate in this range. Some manufacturers use the millinewton-metre instead, which is a tenth of a newton-centimetre, and both conventions appear in the same catalogue depending on which engineer wrote the page.
The exact factor
The factor is exactly 100. The related small units line up as follows: 1 N·m is 100 N·cm is 1,000 mN·m. In American units, 1 N·cm is 0.0885 lbf·in, so a motor rated at 50 N·cm produces 4.4 pound-inches. Micromotors and haptic actuators are often specified in millinewton-metres, where a phone vibration motor produces a fraction of one — the torque scales involved span six orders of magnitude from a phone to a truck driveshaft.
Where you meet this conversion
Small motor and actuator specifications, where datasheets give stall torque and rated torque in newton-centimetres. Stepper motors for 3D printers and CNC machines are usually rated in newton-centimetres or the equivalent in ounce-inches, which is a third convention. Medical device design, watchmaking and instrument engineering use the unit throughout. Screwdriver torque limiters for electronics assembly are calibrated in newton-centimetres.