They're musicians' tools. Playing live music is absolutely destructive to your hearing, and ear protection is required. IEMs are a combination of protective equipment (many are designed the same way as concert earplugs) and earphone, so you can hear a clear signal from the mixing board at a safe volume, while shutting out the house speakers' insane noise levels.

Before IEMs were common, the norm was (hopefully) ear plugs and having monitoring speakers pointed at the musicians from the floor.

IEMs can be made in the concert earplug style, which is usually a center pair with 3-4 layers of rubber domes like typical earbuds. Or you can pay more and have custom molds made of your ears, so ones can be made to perfectly fit.

"Monitor" also has certain implications in audio. Usually something designed for monitoring will have a flat frequency response, while most consumer gear has aggressive EQ curves that do things like boosting bass. (Most people will think monitors "sound bad," while the lack of sound tampering is essential for performance and mixing.)