Adaptations are a variety of mechanisms by which the Borg defend themselves from threats.
Adaptation is generally referred to in the context of shields, in particular, Borg Cube shields.
The Borg can alter their shields to provide more defense against phasers and other weapons. Specifically, the Borg can identify the operating frequency of the beams used against the cube and screen against that frequency, negating all damage from even the full power of a Galaxy-class starship. This makes Borg cubes highly resistant to the weapons of most powers in the Milky Way. The Borg use similar principles to protect individual drones from hand phasers and similar small arms.
Shield adaptation will not negate an unlimited amount of firepower. With a sufficient number of attacking ships, Borg cubes can still be damaged despite their defensive adaptations. Borg defenses often have little effect on weapons they have not previously encountered.
The most common meaning of adaptation in Star Trek is with frequency-based weapons and their affect on shields. Frequency-based weapons such as phasers become ineffective when the Borg have adapted. In combat, Federation starships rapidly change the frequency of their weapons to hinder adaptation. However, rapidly rotating frequencies is, at best, a delaying tactic. Federation hand phasers can also be programmed to rapidly change frequencies, allowing a security officer to disable or kill several drones before the Borg adapt and render his weapon useless.
Adaptation appears in TNG "Q Who" as a formidable self-repair mechanism. After being hit by a phaser fire, the Borg Cube repairs itself. The cube's outer hull is constructed of ductile material conducive to self-repair. It is unclear whether all Borg cubes possess this ability. Other Borg cubes have not shown this ability for rapid repair.