Yes, aquatic animals can and do get diseases. In fact, valuable fish (either as food, or ornamental fish) can even get vaccines – for example, there’s a vaccine for the herpes virus in koi carp. The diseases that we know most about tend to be the ones found in aquariums, as they’re easier to study. I don’t know enough about them to go into detail, but aquatic animals can catch diseases caused by viruses, bacteria, fungus and parasites just like us – although their immune system is a bit different to ours for dealing with them. Fish with parasites on the outside of their body can visit ‘cleaner fish’, which are fish that act a bit like a car wash to remove the parasites.
After that as Bethany and Ramya have said aquatic animals get disease just like all the rest of us. It is interesting that Bethany mentioned Herpes as this virus is also infecting Sea lions in California and biologists in the area are very worried about it.
In addition people that keep fish sometimes need to treat them for various different diseases in peoples fish tank at home fungal disease is especially common and can be a real problem, thankfully most can be cured by simply adding a supplement to the aquariums water.