In modern cryptography, symmetric key ciphers are generally divided into stream ciphers and block ciphers. Block ciphers operate on a fixed length string of bits. The length of this bit string is the block size. Both the input (plaintext) and output (ciphertext) are the same length; the output cannot be shorter than the input — this follows logically from the Pigeonhole principle and the fact that the cipher must be reversible — and it is undesirable for the output to be longer than the input.
Until the announcement of NIST's AES contest, the majority of block ciphers followed the example of the DES in using a block size of 64 bits (8 bytes). However the Birthday paradox tells us that after accumulating a number of blocks equal to the square root of the total number possible, there will be an approximately 50% chance of two or more being the same, which would start to leak information about the message contents. Thus even when used with a proper encryption mode (e.g. CBC or OFB), only 232 x 8 B = 32 GB of data can be safely sent under one key. In practice a greater margin of security is desired, restricting a single key to the encryption of much less data - say a few hundred megabytes. Once that seemed like a fair amount of data, but today it is easily exceeded. If the cipher mode does not properly randomise the input, the limit is even lower.
Consequently AES candidates were required to support a block length of 128 bits (16 bytes). This should be acceptable for up to 264 x 16 B = 256 Exabytes of data, and should suffice for quite a few years to come. The winner of the AES contest, Rijndael, supports key sizes of 128, 192, and 256 bits, but the block size is always 128 bits. The extra key sizes were not adopted by the AES standard.
There are techniques for creating ciphers on unusual or fractional block sizes (i.e. domains whose size is not a power of two); see Format-Preserving Encryption.