Learn German: Der Pfeffer ist scharf.

It refers to the spice, like black or white pepper. If you want to talk about the vegetable (a bell pepper), you would use the word die Paprika. For a hot chili pepper, you use die Chili.

I thought scharf meant "sharp", like a knife. Does it mean "spicy" too?

Yes! In German, scharf is the standard word used to describe spicy or hot food. It can also mean "sharp" when talking about a blade, or "in focus" when talking about a photograph.

Why doesn't scharf have an adjective ending here?

Because the adjective comes after the verb ist rather than directly in front of a noun. In German, adjectives only get special endings when they are placed right next to the noun they describe, such as in der scharfe Pfeffer (the spicy pepper).