Recently I saw an answer about keeping clownfish together that mentioned they were aggressive towards conspecifics. I haven't heard this term before so I don't know what it means. What is a conspecific?
Conspecific is a description of the biological specificity of a species (Biological specificity is a term for studying the interactions between species and the tendencies for behaviours to arise). The two main descriptions for biological specificity are: Interspecific and intraspecific.
If you know what inter and intra mean, then you should be able to guess that interspecificity measures the interactions between one species and another, different, species. On the other hand, intraspecificity measures the interactions between different groups of the same species.
If you want to describe the interactions between individuals, rather than the larger groups, than you can further describe them with the terms heterospecific, or conspecific.
So heterospecificity describes the interactions between two (possibly more) individuals of different species. While conspecificity describes the interactions between two (or more) individuals of the same species.
Taking what we now know about biological specificity, saying that clownfish are aggressive towards conspecifics means that clownfish will be aggressive towards other clownfish, but not necessarily towards other species of fish.
As a side note: In the fish trade, these terms are used a bit more loosely. In most cases a fish that is aggressive towards conspecifics will also be aggressive towards fish that look similar to their species. Some fish will become more aggressive based on certain traits their species has. It all depends on the fish for which characteristics bring out the aggressive behaviour (i.e. body shape or color).