Symmetric-key algorithms is a method in cryptography. It is when the keys for decryption and encryption are exactly the same shared secret. You can generate the secret randomly, or from a password, or through a secret key-exchange procedure like Diffie-Hellman.
Symmetric-key algorithms are very important because they are faster on computers than the other kind:public-key algorithms. In public-key cryptography (asymmetric-key cryptography) the key for encryption can be given to the public with no problem, and everyone can send you secret messages. The key for encryption is "open" because, in practice, it cannot be used to get the key for decryption. This is very useful, but public-key cryptography algorithms are very slow on computers, so they are only used to send a secret key. Then symmetric-key algorithms are used for everything else because they are faster.
There are two kinds of symmetric-key algorithms, called stream ciphers and block ciphers. Stream ciphers encrypt a message as a stream of bits one at a time. Block ciphers take blocks of bits, encrypt them as a single unit, and sometimes use the answer later too. Blocks of 64 bits have been commonly used; though modern ciphers like the Advanced Encryption Standard use 128-bit blocks.
Examples of popular symmetric cyphers include Twofish, Serpent, AES (aka Rijndael), Blowfish, CAST5, RC4, TDES, and IDEA.
In history, some cryptanalysis methods exploited symmetry, so symmetric systems were less secure. Some attacks are called known-plaintext attacks, chosen plaintext attacks, differential cryptanalysis and linear cryptanalysis.
Other terms for symmetric-key encryption are secret-key, single-key, shared-key, one-key and eventually private-key encryption. This last term does not have the same meaning that the term private key has in public-key cryptography.