The common parasites that affect Koi can be classified into 2 types: ones that can be seen with the naked eye and ones that can't. Parasite's usually don't kill a Koi, but can lead to more serious ailments like bacterial infections.
Parasites that can be seen with the naked eye:
Fish Louse (Argulus)
Looks like a small circle with two dots. Argulus usually hangs out on the underside of Koi
Anchor Worm (Lernea)
Can be seen attached to the side of a Koi. Looks like a piece of thread.
Ich (Ichthyophthirius multifilis)
Also know as "white spot", Ich looks like little pieces of salt.
Parasites that can't be seen with the naked eye:
Looks like a spinning flying saucer. Trichodina causes web-like lesions.
Flukes (Gyrodactylus & Dactylogyrus)
Flukes usually hang out on the skin and gills of a Koi. The main difference between Gyrodactylus and Dactylogyrus is that Gyrodactylus is a live bearer and Dactylogyrus lays eggs.
Costia (Ichthyobodo Necatrix)
Costia is one of the smallest parasites that affect Koi. They are hard to find, you will need a microscope that has at least 40X magnification to find Costia.