As explained in a [comment][1], a generic attack is one that works against all block-ciphers (with a given block and key size), without consideration about the structure of the block-cipher.

One generic attack for a block cipher of a given block size $b$ bits is one that, for a fixed key, builds a _dictionary_ of input/output pairs (e.g. from past plaintext/ciphertext). When an input or output in that dictionary gets reused, the adversary gains an advantage. In many scenarios, that's expected after about $2^{b/2}$ blocks (less in ECB mode, more in CTR mode). Such attack works for any block cipher, including an hypothetical one implemented as a random permutation.

If we in addition consider the key size of $k$ bits, another generic attack, _brute force key search_, enumerates the keys. With at least $k/b+1$ input/output pairs, that's likely to find the key after about $2^{k-1}$ attempts.

Sometime we have a generic attack against a whole category of block ciphers sharing a common characteristic. For example, there's a generic attack against all Feistel Ciphers, based on the fact that for any key, they implement an [even permutation][2]. This allows an adversary having built a dictionary of all input/output pairs except two of these, to deduce the remaining two with certainty.


  [1]: http://crypto.stackexchange.com/questions/14547/block-ciphers-and-non-generic-attacks#comment29428_14547
  [2]: http://en.wikipedia.org/wiki/Parity_of_a_permutation