Something permanent stays forever, or lasts forever. Or, it doesn't change, or you can't change it.
For example, if you have a permanent home, or a permanent address, that's great! It means you live in one home, with no plans to ever move. And if you give a pet a permanent home, it means you give it a forever home: you plan to keep your pet and love it and take care of it for its entire life.
And, compared to temporary or impermanent hair dye that changes your hair color for just a little while, permanent hair dye changes your hair color for good: it stays that color, and it won't wash out.
So far, permanence seems like a pretty good thing, right? Your love for your family is permanent; it doesn't ever stop. Your teeth are permanent; they'll last you all your life. (Probably.) But permanence isn't always good.
For example, a permanent stain will stay forever: you can't clean it off.