The Turkic nation-states that exist today are Turkey, Azerbaijan, Turkmenistan, Uzbekistan, Kyrgyzstan and Kazakhstan. There are also the Crimean Tatars, Tatars in other parts of what is now Russia and eastern Europe, the Bashkirs in Bashkortostan (Russia), the Uighurs in Xinjiang (China), and some other smaller Turkic groups in places like the Caucasus. That covers a pretty massive geographical area so you might have to be more specific with your question.

Regarding the Turks of Turkey and Azerbaijan (and also Turkmenistan, and also the Turkmen of the Arab world), their origin lies in the migration of the Oghuz, a Turkish tribe, mainly from the region north of Lake Aral (in modern Kazakhstan), into other parts of Central Asia (e.g. Transoxiana) and the Middle East. These migrations began in late 10th when the collapse of the Samanid state in Central Asia, an Iranian Sunni polity, created a political vacuum.

If you're interested, here's a short academic encyclopedia entry on the subject: https://www.academia.edu/1514276/Oghuz_Turkish_Tribe

By Turkish people, do you mean the modern state of Turkey, or generally the Turkics from central Asia?

There were several closely connected Tribes lived in Central asia and modern Mongolia. Think like todeys provinces as "tent towns", there were several tent towns cities etc. They all named differently as their population(like village,town,city) . But difference between them and cities "they were at move" .

By the time Turks migrated different places, mixed with different peoples, formed different federations and created nowadays Turkic peoples.