Both terms refer to a choice involving two options. Let's say you have option A and option B.
To choose EITHER A or B means you must choose one; you cannot have both. If you choose A you cannot also have B.
To choose NEITHER means that you do not choose Aor B. You refuse both of them.
If you're a math or logic person, either allows us to use the "Exclusive Or," in everyday speech; some might call it XOR. If I could "choose A or B," I would have the option to choose just A, just B, or both A and B.