Any dictionary will explain the difference quite clearly I reckon, but here's my two cents in a nutshell:
To ADOPT : to take, to choose, to select
* The Senate adopted the report of the Commission. (they accepted it, by voting in favor of it)
* Jack and Jill have adopted a Tsunami orphan.
* The young boy quickly adopted a new identity.
To ADAPT : to adjust, to change, to rework, to modify
* The Senate adopted the report, although some senators would have preferred to adapt it before voting on it.
* The adopted Tsunami orphan quickly adapted to our Western values and now he's even adopted a new nickname 'Tsunidude'.
* His parents had quite some problems adapting to the new identity their son had adopted.
Adapt means to change something you do so you can be more comfortable.
For example: People adapt to the weather.
Adopt means to take something as yours.
For example: Many American families adopt children.