Different cultures had different methods. At the university I went to there was a social anthropologist studying tattoos.
Usually a sharp needle would be used (or something like it) and for ink, usually something based on soot. Still practiced, as a stick and poke (with sewing needles) or Thai fish bone tattoos.
A different way was to pull thread and needle through your skin with pigments on the thread, that produces straight lines and is extremely painful.
Yet another way was to make a cut/incision on your skin and rub pigments directly in.