All four words describe something that can be changed, but they differ in context and scope. 'Customizable' implies alteration to suit personal taste or needs — common in product descriptions and software interfaces ('a customizable toolbar'). 'Adjustable' focuses on physical or mechanical changes of position or degree ('an adjustable chair'). 'Configurable' is more technical, used for systems where components can be arranged in different ways ('configurable network settings'). 'Adaptable' is broader, describing something that can function well under different conditions ('an adaptable plant that grows in any soil'). Choose 'customizable' when the change is driven by individual preference; use 'adjustable' for physical settings; prefer 'configurable' in technical contexts; and pick 'adaptable' for general flexibility.
Formed by adding '-able' (meaning 'capable of being') to the verb 'customize', which was created in the 20th century from 'custom' (from Old French 'costume', meaning 'habitual practice') plus the verb suffix '-ize'. The idea of making something according to a customer's individual wishes grew out of the broader sense of 'custom' as something a person usually does or prefers.