Which are the most  common  endocrine  skin   diseases ?

Endocrine diseases are those affecting the glands producing hormones (adrenal glands, thyroid, ovaries, testicles etc.).

Those with skin symptoms in cats are very rare while in dogs the most frequent are hyperadrenocorticism (Cushing's disease) and hypothyroidism.

The first is due to an excess of cortisone in the blood caused by tumors of the pituitary or adrenal glands or by excessive administration of corticosteroidal drugs (iatrogenic Cushing). 
It's more common in small breeds. The skin thins, the hair on the trunk falls, the abdomen enlarges; dog is eating and drinking very much and breathes panting.

In hypothyroidism instead the thyroid is damaged by a chronic inflammation and the production of thyroid hormones is reduced. More common in large breeds, also in this case there is hairloss and skin is often hyperpigmented and undergoes excessive scaling; the dog tends to gain weight and become lazy.

Note that both diseases are not involving only the skin but are systemic (ie also involve internal organs).

Diagnosis is made primarily with blood tests and through diagnostic imaging (eg. ultrasound).
The prognosis is good for hypothyroidism, guarded for hypercorticosurrenalism.

During therapy (which generally lasts for the whole doglife) it is important to have regular blood checks.

Iatrogenic Cushing heals suspending corticosteroid therapy.   Adrenal tumors can be removed surgically.