Which condition primarily affects the eyelids and can lead to discomfort when an oil gland is infected?

The condition that primarily affects the eyelids and is associated with discomfort due to an infected oil gland is a hordeolum, commonly known as a stye. A hordeolum is an acute infection of the oil glands located at the base of the eyelashes, which can lead to inflammation and swelling of the eyelid, causing pain and tenderness.

In contrast, a chalazion is a blockage of an oil gland that occurs without infection, leading to a slow-growing lump on the eyelid that may become inflamed but is not typically associated with immediate discomfort from infection.

Conjunctivitis affects the conjunctiva, leading to redness and irritation of the eye itself rather than the eyelids specifically, and does not involve infected oil glands. Blepharitis is an inflammation of the eyelid margins, which can be chronic and may produce discomfort, but it primarily involves inflammation rather than the acute infection of oil glands.

Therefore, the hordeolum is the condition that is directly linked to the discomfort when an oil gland in the eyelid becomes infected.