"Forex" is a market (Foreign Exchange).
"Day Trading" is a style of trading in which you enter and exit a market position in the same day.
"Position Trading" involves maintaining a market position, or trade, over an extended period of time (weeks or months).
"Swing Trading" falls in-between. It involves catching the back-and-forth movements over an weekly/monthly period.
Any of these trading styles can be used in FOREX.