Does first name come first name or the last name?

That depends on several things. In America, the first name is placed first and then the middle and last name follow. My name, for example, is Donna Lee Halper (or just Donna Halper, or Donna L. Halper). But in some Asian countries, like China and Korea, the last name comes first. So, the woman who is the President of South Korea is Park Geun-hye; her last name comes first, and she would be referred to as President Park.

Back to English, and this is also the same rule for languages like French, Spanish, and Italian. Under normal circumstances, it's your first name, sometimes your middle name or middle initial, and then your last name. But there are some exceptions, most notably in research papers when you are writing your bibliography. Bibliographies require last name and first name, separated by a comma. So, my bibliographic entry would be Halper, Donna Lee (or Halper, Donna L.).