What is a internal hordeolum and a stye? same thing?

Not the same: A stye generally refers to an external hordeolum, which is an infected hair follicle. These are painful enlarge into a red, round bump on the edge of the eyelid. An internal hordeolum is a chalazion or inflamed, backed up oil gland in the eyelid margin. They form a firm, painless nodule deep in the eyelid and generally are treated with hot compresses and lid scrubs.

Clogged oil gland: A stye is a clogged oil gland found in the eyelid margin, it produces oil for the surface of the eye. When the orifice of this gland becomes blocked, the oil will back up and form what is called a stye. Typically a stye is from inflammation causing the opening (orifice) to occlude causing the stye. A stye is typically not an infected oil gland, but can occur due bacteria causing inflammation.