What are the levels of organization within the biosphere?
From largest to smallest: biosphere, biome, ecosystem, community, population, and organism.
Going from the largest, most inclusive to the smaller, most exclusive we have:
*You may want to review the definition of a species before reading further.
A population is a group of individuals within the same species, within the same area, and that are capable of reproducing with one another.
A community is a defined as the populations of two or more species within the same geographical area and within the same time period.
An ecosystem is a community along with its abiotic (non-living) components, such as the water and air.
A biome is a large ecological area with similar climatic conditions. A biome can have multiple ecosystems within it.
The biosphere includes all living organisms along with the spheres they interact with (lithosphere, hydrosphere, etc.).
See these related questions for more information:
What is the difference between a population, a community, and an ecosystem?
How does an ecosystem differ from a biome?