The Turkic peoples is the term generally used to refer to all peoples speaking Turkic languages (which encompasses a broad variety of ethnic types). Turks, or Turkish people, is generally used to refer to the Turkic people who became part of the Ottoman Empire, and later the country of Turkey.

The Huns may have been Turkic people, but this is unknown. Some think they were more likely to have been speakers of an Indo European Language. Of course, the Huns were not the people who are now Hungarians.

The Hungarian language, which is spoken by the Magyars who now live in Hungary, is a Finno-Ugric language, not a Turkic language. So, the people who now live in Hungary are not a Turkic people.

The Turks in Turkey, who did take over Constantinople (which is now Istanbul) are a Turkic people who migrated into the Middle East from somewhere in Central Asia.

Turkmenistan is primarily inhabited by Turkic peoples, as is much of central Asia. The Turkic people are first referenced (by the Chinese) however, in what is now Mongolia. They may have originated there rather than in Turkmenistan. The Mongolian language, currently spoken in Mongolia, however, is in the Mongolic language family, not the Turkic family. If the Turkic people did first come from Mongolia, they left there at some point and were replaced by Mongolic speakers.