Ecology uses three closely related terms that are easy to blur together but actually describe genuinely different scales. An ecosystem is a specific, local community of interacting organisms together with their immediate physical environment — a particular pond, forest or coral reef. A biome is a much broader category, a type of large-scale ecological zone, like tropical rainforest or temperate grassland, defined mainly by shared climate and characteristic vegetation, of which many individual, separately located ecosystems can be examples. The biosphere is broader still: the entire global layer, spanning every biome, within which life exists on Earth.
An ecosystem is a specific, bounded community in a particular place
An ecosystem refers to a genuinely specific, localised set of interacting organisms together with the physical environment they inhabit and depend on — a particular lake, a specific patch of forest, one coral reef. Ecosystems can be defined at quite different physical scales themselves, from a small pond to an entire watershed, but what makes something an ecosystem in the ecological sense is that it names an actual, located community of organisms interacting with each other and with their specific physical surroundings, not a general category of place that could describe many separate locations at once.
A biome groups many separate ecosystems that share a climate type
A biome, by contrast, is a broad classificatory category defined chiefly by climate and the characteristic type of vegetation that climate supports — tropical rainforest, temperate grassland, tundra, desert. A single named biome, like tropical rainforest, actually encompasses many separate, individually located ecosystems around the world, in South America, Central Africa and Southeast Asia alike, all sharing the same broad climate-defined biome classification while still each being their own specific, distinct ecosystem with its own particular species and local conditions. The biosphere sits above both, naming the entire global layer across every biome and every individual ecosystem within which life on Earth actually exists, from the deep ocean floor to the upper atmosphere wherever living organisms are found.
What we're still unsure about
These three terms' definitions, and the fact that they describe genuinely different scales rather than being interchangeable synonyms, are well established, standard ecological terminology. What's more genuinely a matter of ongoing scientific judgement is exactly where to draw a given ecosystem's own physical boundaries in specific real-world cases, since ecosystems don't usually have sharp, self-evident edges the way the term's tidy textbook definition might suggest — ecologists studying a particular location often have to make a reasoned, sometimes genuinely contestable judgement call about where one ecosystem's boundary actually lies and where a neighbouring one begins, rather than there being one single correct answer built into nature itself.
This sits inside Ecosystems, Biomes & Biosphere, one of seven topics in Ecology, one of six domains in Biology, one of seventeen subjects the app can quiz you on.