Anything that doesn't alter or shift over time is unchangeable, like the unchangeable rules at your school or your unchangeable conviction that pistachio is the tastiest flavor of ice cream.
Things that can't be changed are unchangeable, and this adjective tends to be used for describing really big ideas, like the unchangeable laws of the universe or the unchangeable nature of love. You can use it for anything that seems steadfast, firm, or permanent, like your unchangeable affection for your best friend or your sister's unchangeable insistence that she has to ride in the front seat.