'Adrenal Fatigue' is more of a marketing term than anything, to refer to a variety of symptoms associated with underactive set of adrenal glands. This condition, medically referred to as 'Adrenal Insufficiency', is usually when the adrenal glands become hyporesponsive (underresponsive) to signals sent from the brain to secrete hormones. This, however, is not solely to place blame on the adrenal glands; abnormal effects can occur at a variety of points.
Mechanistically, the Hypothalamus (a brain structure) perceives stress and secretes Corticotropin Secreting Hormone (CRH) which acts on the pituitary. The pituitary then releases Adrenocorticotropin Hormone (ACTH) which goes to act upon the adrenal glands, which rest on top of the kidneys. The interaction of these three organs is known as the Hypothalamus-Pituitary-Adrenal axis, or the HPA Axis.
Primary Adrenal insufficiency is when the adrenal glands themselves respond less to secreted ACTH or are otherwise damaged, and despite the hypothalamus and pituitary working fine the adrenal glands do not secrete the hormones required of them; cortisol and Dehydroepiandrosterone. This can be diagnosed with an ACTH-stimulation test by an endocrinologist, and the main pathology is less secretion and synthesis of cortisol.
Secondary Adrenal insufficiency is either a hypothalamus that secretes less CRH, or a pituitary that secretes less ACTH, in response to stress. The adrenal glands themselves function normally in this scenario, but do not receive the signal to create more hormones. ACTH can be measured in the blood by a medical doctor, but CRH is difficult to measure practically due to it being localized in the brain.
Sometimes, deficiency is CRH is referred to as tertiary adrenal insufficiency.
Thus, 'adrenal insufficiency' is a dysregulation in the HPA axis that ultimately results in less secretion of hormones that can regulate mood and energy. It really isn't one mechanism or event, but a term used to refer to the ultimate state of less hormones and dysregulation in this system