Cast Iron

First you remove the old seasoning using one of a few methods. That leaves you with bare iron. Then you re-season.

My preferred way to strip a pan of its old seasoning is to heat the pan up to 200F in the oven, then take it outside and spray it with oven cleaner. Then I put it in a trash bag and bring it inside (or leave it outside if it's summer) and let the oven cleaner do its thing. Rinse and repeat until there is no more seasoning. Wear protective equipment, of course.

Other methods include an electrolysis bath, or using the self-clean cycle on your oven (It's possible to damage the pan this way, but chances are it'll be fine).

To re-season, I like to use Crisco.

1. Warm the pan to 200F in the oven
2. Remove and apply a coat of Crisco. Be thorough.
3. Remove excess so you have only a very thin coat of oil
4. Bring the pan to 300F in the oven for 15 or so minutes
5. Remove and give the pan an additional wipe down of any excess oil that might have pooled during the 300F cycle
6. Bake upside down at ~460F (some people do 475F) for 2 hours. The smoking point is above 500F so we leave it just below that to polymerize.
7. Turn oven off, but leave the pan in while the oven cools down (takes hours).

I do that cycle once a day until the pan looks good. It takes anywhere from 4 to 80 coats. Just kidding about 80, but someone on reddit did that.

The pan will NOT be shiny after seasoning. Seasoning is a permanent layer of polymerized oil. You still need to cook with your favorite cooking oil, which will make the pan shiny.

Washing the pan with soap will leave it looking dull again, because you've washed away the cooking oil. The seasoning will still be intact unless you've destroyed it some other way (tomatoes are acidic and can wear down the seasoning, putting it in the dishwasher will destroy the seasoning, etc.)