Learn French: Cette mauvaise histoire est drôle.

You are right that most French adjectives go after the noun. However, mauvais (bad) is an exception. It belongs to a small group of common, short adjectives describing goodness, beauty, age, or size that go before the noun (just like bon, petit, and grand).

Why do we use cette here instead of ce or cet?

Cette is the feminine word for "this" or "that", and we use it because histoire is a feminine noun. You might remember that we use cet before words starting with a vowel or silent 'h', but that rule is only for masculine nouns. For feminine nouns, it is always cette, no matter what letter the noun starts with.

Why does mauvaise end in an -e, but drôle does not seem to have a feminine ending?

Both adjectives are matching the feminine noun histoire. The masculine word mauvais gets an extra -e to become the feminine mauvaise. However, the base dictionary form of drôle already ends in an -e. When a masculine adjective already ends in an -e, it does not change for the feminine form.

I have seen histoire translated as "history". How do I know when it means "story"?