Algebraic attacks are a class of techniques which rely for their success on some block cipher exhibiting a high degree of mathematical structure.
For instance, it is conceivable that a block cipher might exhibit what is termed a group structure. If this were the case, then encrypting a plaintext under one key and then encrypting the result under another key would always be equivalent to single encryption under some other single key. If so, then the block cipher would be considerably weaker, and the use of multiple encryption (see Question 85) would offer no additional security over single encryption; see [KRS88] for a more complete discussion. For most block ciphers, the question of whether they form a group is still open. For DES (see Question 64), however, it is known that the cipher is not a group.
There are a variety of other concerns with regards to algebraic attacks. See [Rob95a] for more details.