Mid-bosses (also known as mini-bosses) are enemy characters that are stronger than regular enemies but not as strong as bosses. Kirby usually has to fight them in the middle of a stage before he can continue on (which is probably where they get their names from). Usually, during a fight with a mid-boss, Kirby cannot leave the battlefield and thus must fight them to progress.
Most mid-bosses, with the exception being the Meta-Knights have a health meter that must be depleted before they are defeated. They also have at least one attack that can be inhaled and spit back at them (this is the only way to beat them without a copy ability).
Copy abilities from mid-bosses
When a mid-boss is defeated, it normally lies on the ground for a few seconds before exploding (except for Sphere Doomers, Kracko Jr., Lololo, Super Bonkers, all DX mid-bosses, mid-bosses in Kirby 64: The Crystal Shards, Kirby's Epic Yarn, and Kirby Mass Attack). During this time, Kirby can inhale them and gain their ability. Nearly all mid-bosses have copy abilities and some copy abilities can only be collected from a certain mid-boss (e.g. the Ghost Ability).