Technically, a predator-prey relationship is one type of symbiosis. Symbiosis is any type of interaction between two different species of living things in the same environment. A predator-prey relationship is between two animal species —one kills and eats the other. Not all sources include this as a type of symbiosis, arguing it is different from the three other types of symbiotic interactions between organisms.
In the first of these, mutualism, both species involved benefit in some way. An example of mutualism is a bee visiting a flower — the bee obtains food, and the plant is pollinated.
Commensalism is an interaction benefiting one organism, and neither benefiting nor harming the other. Barnacles attach to large marine animals such as whales to move to where food is found; the whales are not affected.
Parasitism involves a parasite living on or in an organism. The parasite benefits, and the host is negatively affected, but usually not killed; a successful parasite does not kill its host so it continues to have a place to live. Fleas and ticks on a dog, or a tapeworm in its digestive tract, are examples of this type of interaction.
The predator-prey relationship can be considered to be a type of symbiosis because it is the interaction of two species. It can also be considered to be similar to parasitism, as one organism benefits while the other is negatively impacted. The predator-prey relationship can be considered be different from the other types of symbiosis since one of the organisms does not survive the interaction.