A multi-target attack is one in which it is attacked many keys simultaneously, and the adversary is content with success for any of the keys. That might allow increased chance of success at a given effort, sometime linearly with the number of keys.
For example, for the block cipher DES, if it's known $P$ and $\operatorname{ENC}_{K_i}(P)=C_i$ for $k$ random keys $K_i$, there's a simple speedup by a factor of nearly $k$ (for $k$ up to some thousands at least), where an adversary encrypts $P$ under incremental keys and searches the resulting $C$ among the $C_i$ (the search can be arranged to cost essentially one memory access).
In a public key context, the adversary may have many public keys, and be trying to find one private key.
