In cryptography you often use hybrid system. That is a combination of asymmetric and symmetric techniques.
The asymmetric key exchange: An example for that is Diffie-Hellman. A key exchange is important in situations, where you have to find a secret key using a public way to exchange informations. A symmetric key exchange is not possible, so you need to use an asymmetric one.
The symmetric key encryption/decryption: Asymmetric system are slower than symmetric systems. So you use symmetric encryption/decryption to make sure no one, who does not have the key, can read your messages. (or at least have to do a lot of work).
How the systems work is different for each system. The security often depens on different mathematical problems. E.g. RSA and Diffie-Hellman are used for key exchange. RSA is based on the factorization problem, Diffie-Hellman is based on the discrete logarithm problem. This means, the way the data gets encrypt/decrypt is different, too.