What are the three main types of cryptographic functions and how do they differ?

Symmetric uses one shared key (fast); asymmetric uses a public/private key pair (solves key exchange); hashing is one-way and keyless (irreversible).

Why all three exist: Symmetric is fast but requires secure key exchange. Asymmetric solves key exchange but is slow. In practice, TLS uses asymmetric to exchange a symmetric key, then symmetric for the actual data (hybrid approach). Hashing is fundamentally different — it's irreversible, which is exactly what you want for passwords.