Literally, 'becoming bare', from the Greek root psîlo- meaning 'bare'. It has two senses in pathology and one in phonetics.
First, simply hair loss; and secondly, an old word for sprue, a disease characterized by loss of the epithelium of the intestine and consequent weakening of ability to digest. We need some other expert to tell you about those.
In language, it means the loss of initial [h], the aspirate. It usually refers to the process in some dialects of Classical Greek, and then in all forms leading to Modern Greek. A synonym is deaspiration.
In Classical Greek the h sound only occurred at the beginnings of words. It usually came from a proto-Indo-European s: compare Greek hemi-, helio- with Latin semi-, sol. But in some cases it came from original y, w. In the Aeolic dialects psilosis had taken place even before the classical period: e.g. in the Lesbian dialect used by the poets Sappho and Anacreon. Varieties with psilosis are called psilotic.
In near-classical times the Ionic dialects of Asia Minor lost their h. It spread, and is recorded in the Koine, the colloquial Greek that was to give rise to New Testament Greek and the modern language, in the 3rd century BCE. This also left its mark on the alphabet. The Semitic consonant heth was borrowed in the form H to represent the sound it still has in English. But the eastern dialects didn't have an H sound, so they reused it as a vowel. In a spelling reform of around 400 BCE the western dialects, including the Attic dialect of Athens, adopted the eastern convention. Then they had no letter for the initial h they still pronounced, so broke a small H in half and used the bits: see breathing marks in Greek for the full story.