What type of relationship does a cat and tapeworm have?

Parasitism.

Explanation:

Parasitism is an association in which one organism, lives on another, the host, deriving benefit from it and causing harm to it.

The tapeworm here is a parasite and it lives inside the cat's (host) body where the tapeworm is specialized for its mode of life. It survive and reproduces very well without needing another specie because it is a hermaphrodite. It causes loss or poor development of unwanted organs such as the digestive, sense organs and muscles. Its long flattened body also provides a large surface area for absorption of nutrients too.