I'm a little confused on how these terms are thrown around in a book that I am reading. Are they all different terms that refer to the same concept, or are they slightly different implementations corresponding to the same thing but in different types of filesystems? Any input is appreciated :)
They're completely different things:
A 'file header' is a part of a file that identifies the contents of the file. Some file types specify a file header, some don't. Most document, image, and video formats have a file header from which the file type and basic file parameters can be identified.
A 'bit map' keeps track of what space is used and what space is free. It's usually used by a filesystem to track the usage of space on a volume.
An 'inode' contains file system metadata needed for the filesystem to find the file, know how large it is, and know which blocks contain the data in the file.