This type of exocrine gland is a simple, branched acinar gland connected to a hair follicle.

I believe the answer is Sebaceous gland. Sebaceous glands are microscopic exocrine glands (glands that secrete and produce substances using ducts to an epithelial surface). They are found in the skin and secrete oily or waxy substance called sebum. Sebum functions to lubricate and waterproof the skin and hair of mammals.