Remember Black Beauty and the issue surrounding bearing reins. A bearing rein is sometimes called an overcheck or checkrein.
The purpose of the bearing rein or overcheck rein on harness horses was to elevate a carriage horse's head - and at the time they were often used in a manner that was uncomfortable for the horse and made it hard for them to lower their head enough to pull properly. Bearing reins in that form disappeared...because of the novel (These days, we think of Black Beauty as a cool kids' book, but it was written to highlight the abusive practices and fashions of the time).
Overcheck reins are still used on some harness horses, but are fitted much more loosely. They're most often seen on Hackneys and other horses with naturally high head carriages. Some believe they help prevent the reins from being caught in the shafts (but if they're holding the horse's head that high, they're probably too tight.
The other context in which you might hear "overcheck" is with animals that are used to carry small children or disabled riders. A loose overcheck rein is often fitted to these horses with the primary purpose being to stop the animal from putting its head down to eat - which with a light or insecure rider can result in the rider being pulled over the head by the reins (don't laugh. I've seen it happen). In this case, the overcheck or bearing rein is fitted very loosely and not intended to interfere in any way with how the horse holds its head.
In this picture of a show Hackney, the bearing rein is the strip of leather that appears to be running along the top of the neck. Source: John Goetzinger via Wikimedia Commons.