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In mutualism, two species interact in a way that is beneficial to both. An example is cleaner fish found around coral reefs - where a small fish eats food debris, dead skin, and parasites off of a larger fish, including within their mouths and between their teeth. The small fish gets food while the larger fish gets healthcare.
Commensalism is where one species benefits from associating with another, but neither helps nor harms the other. For example: birds that follow large grazing mammals in a grassland, which stir up insects as they move about, which the birds then eat.
Parasite species benefit from associating with another species while actively harming them, but without (intentionally) killing them. For example, tapeworms live inside the digestive tract of vertebrates, absorbing some of the nutrients from food that the vertebrate eats. Mosquitoes and fleas ingest blood from mammals and birds. Cuckoos and cowbirds lay their eggs in the nests of other bird species, often displacing the chicks of the other species in the process.
Finally, one species can invade (or infect) the body of another to cause disease. The infecting species can be a parasite if it's some sort of multicellular eukaryote, but is called a pathogen if it's a bacterium, virus, or fungus.