A distinguishing attack is any form of cryptanalysis on data encrypted by a cipher that allows an attacker to distinguish the encrypted data from random data.
To prove that a cryptographic function is safe, it is often compared to a random oracle. If a function would be a random oracle, then an attacker is not able to predict any of the output of the function. If a function is distinguishable from a random oracle, it has non-random properties. That is, there exists a relation between different outputs, or between input and output, which can be used by an attacker for example to find (a part of) the input.
This information might then reveal the encryption method used, some information about the encrypted message, or refine the potential key space.