The problem with symmetric cryptography is how to distribute the secret key to the involved parties. In assymetric cryptography (also: public key cryptography), algorithms use two different keys: a private and a public one. A message encrypted with a private key can be decrypted with its public key (and in some cases vice versa). The owner of the key pair holds the private key, and may distribute the public key to anyone. Someone who wants to send a secret message uses the public key of the intended receiver to encrypt it. Only the receiver holds the private key and can decrypt it.
Compared to secret key encryption, public key encryption is slow.
A popular assymetric cryptographic algorithm is RSA, used in PGP.