If someone's not connected to a certain group or organization, they're unaffiliated. If you vote, but you're not registered as a Democrat or a Republican, you're unaffiliated.
Not being connected or associated with a specific political party is one way to be unaffiliated. If you don't belong to a particular religious group, you can describe yourself as religiously unaffiliated. The origin of this word is the verb affiliate, "join or be closely connected," and its Latin root affiliatus, the past participle of affiliare, "to adopt a son."