Modern Turkish names look a lot like most European names: first name (ad), and surname/family name (soyadı). This is thanks to a 1934 law, creatively called the Law on Family Names, that required all Turks to adopt some kind of surname. Prior to that, Turkish names probably looked a lot like the typical Arab name, where the “surname” may just identify the person via anything from their occupation to their birthplace.
Many Turkish first names still resemble Arab given names, but they are rendered in the modern Turkish (modified Latin) alphabet; back in Ottoman times, when everything was written in Arabic script anyway, they would have been identical to their Arabic counterparts. But modern Turkish Latinizes Arabic words based on how they sound (according to the modern Turkish alphabet) rather than how they’re written in Arabic script, so (for example) the common Arab name “Abd al-Rahman” becomes “Abdurrahman,” and “Jamal” becomes “Cemal.”
Surnames are passed patrilineally, from father to children, and can hearken back to the family’s geographic origins or tribal origins, or really whatever the head of that family picked when the 1934 law went into effect, but may also indicate descent from an important ancestor. Surnames that end in -oğlu and -zade (from Persian) indicate descent from whatever name that particle is added on, though -zade indicates that the ancestor was someone of noble birth whereas -oğlu refers to commoners.
When asking for someone’s name, the question “what is your name?” is adınız ne? and the response is “my name is _____,” benim adım _____, or “I’m _____,” ben _____.