What are spongy organs consisting of prograssively small air passage ways leading to Avleoli?

The structure you are describing are lungs.

Lungs are soft, spongy, and elastic organs in the chest. They are responsible for taking in oxygen and releasing carbon dioxide during the process of breathing. The lungs are made up of smaller structures called bronchi, which divide into smaller branches called bronchioles, which eventually lead to tiny air sacs called alveoli. The alveoli are where the exchange of oxygen and carbon dioxide takes place between the air and the bloodstream.