An 'ecosystem' is an area where organisms depend, compete, and interact with each other.
The term is a concept rather than a strongly defined area or interaction, but usually relates to the organisms within a geographically defined area or dominant feature. A lake, river, coral reef, swamp, or forest could all be described as an ecosystem.
In each of these ecosystems there are organisms that share a food chain and recycle nutrients and compounds. For instance, waste products are broken down by bacteria and microscopic organisms, which release nutrients used by plants, which are eaten by fish, which release waste products, and so the cycle begins again. This is an extremely simplified example of the processes that happen within an ecosystem.
In aquarium terms, 'ecosystem' is often used to relate to the recreation of a particular environment within the aquarium, or to encompass all of the biological process that are carried out in the aquarium or 'aquarium ecosystem'.