Very generally, this is “a devoted member of an organization.”
It has its basis in Russian, where it was used to refer to a devoted member of the Communist Party. The party itself was referred to as “apparat,” meaning something along the lines of “The Communist Machine.”
Today, it’s used to refer to anyone who is a blind or devoted follower of some organization – an apparatchik is a loyal cog in a larger machine.
It’s vaguely pejorative. It implies that the subject doesn’t think for themselves, they just do what the organization tells them to do.
Why I Looked It Up
It came up in a book about the structure of the Soviet Union. Someone was claimed as having “amassed enough apparatchik” to achieve some goal, which means they had gained an amount loyal followers inside the party.