What is the name or concept that explains or refers to when chords become other chords if you move them one step up or down? For example, B♭ Major becomes A Major once it's moved a step down, and vice versa (A major becomes B♭ Major when it's moved a step up). Is this the same as key modulations as well?
The concept you refer to is called Transposition, which is moving one or more notes up or down by a given amount. You can transpose individual chords, or phrases, or entire pieces.
For example, if you have a piece in C major, and transpose it up a whole step, the resulting piece will be in D major. If the piece modulates partway through, for example, from C to G (up a fifth), then the transposed piece will modulate from D to A (also up a fifth).