What baking powder actually is
Baking powder is baking soda (a base) pre-mixed with a powdered acid and a little cornstarch to keep the two apart and dry until they meet moisture. Once wet, the acid and base react and release carbon dioxide — the bubbles that make batter rise. Baking soda alone needs an acidic ingredient already in the recipe (buttermilk, yogurt, brown sugar, cocoa) to do the same thing; baking powder carries its own acid so it works even in recipes with nothing acidic in them. That's exactly why you can rebuild it: combine baking soda with any of the acids below in the right ratio, and you've made your own baking powder on the spot.
Which method to use when
Cream of tartar is the closest match to real baking powder — same dry, neutral-flavored format, just without the cornstarch buffer, so mix it in right before baking rather than ahead of time. Lemon juice or vinegar works nearly as well and is more likely to already be in the kitchen, though it adds a small amount of extra liquid and a very faint flavor. Buttermilk or yogurt is the option to reach for when the recipe already calls for milk or another liquid — swapping some of that liquid for buttermilk or yogurt gets you the acid you need without adding anything extra to the recipe.
A note on double-acting vs. single-acting
Most store-bought baking powder is "double-acting" — it releases some gas as soon as it gets wet, and more again once it hits oven heat, which is more forgiving of batter that sits around before baking. A homemade soda-and-acid mix is "single-acting" — it reacts fully as soon as it's wet, with no second reaction from heat. In practice this just means: mix it in and get the batter into the oven promptly, rather than letting it sit on the counter.
FAQ
Can I just use baking soda instead of baking powder? Not on its own, unless the recipe already has a strongly acidic ingredient in it — baking soda needs an acid to react with, and using it alone (or in the same amount as baking powder) can leave a noticeably bitter, metallic taste.
Does the substitute taste different from real baking powder? Cream of tartar and lemon/vinegar versions are close to undetectable in the final result; buttermilk or yogurt will add a very slight tang, which is usually pleasant in things like pancakes or biscuits and barely noticeable in cakes.
How do I know if my baking powder is still active? Drop about ½ teaspoon into ⅓ cup hot water — fresh baking powder fizzes vigorously right away. Weak or no fizzing means it's lost potency and needs replacing (or use one of the substitutes above in the meantime), which is also a common, overlooked reason a cake comes out flat even when the recipe and amounts were followed exactly.