The small pot used to brew Turkish coffee is called a cezve.
A cezve is a small, long-handled pot with a pouring lip, traditionally made from copper or brass. Its tapered shape helps the coffee develop a layer of foam while it is heated gently. Modern versions may also be made from stainless steel, aluminum, or ceramic.
Turkish coffee is prepared with extremely fine coffee grounds, water, and sometimes sugar. The mixture is heated until it foams, but it is usually removed before a vigorous boil. The unfiltered grounds remain in the cup and settle at the bottom after serving.
The vessel is also called an ibrik in some English-language contexts, although ibrik traditionally means a pitcher or ewer in Turkish. Related coffee pots have different names across the Balkans and Middle East, including džezva, kanaka, and rakwa.