Any type of ballistic body armor meant to be concealed or hidden in plain sight. Covert armor can be worn underneath plain clothes, or installed inside plain-looking attire.
Although there are examples of hard plates worn covertly, in most cases cover armor is soft, made with ballistic fabrics. Outer fabrics can range widely depending on the intended use.
No. Unless you are a convicted felon with no express permission to wear body armor, it is perfectly legal to wear any type of armor in public. Covert armor is the preffered choice for most cases.