Refers to one's overall fortune or luck, often in the context of fortune-telling, horoscopes, or general life outlook.
I had my fortune told for today.
He has strong luck.
I had a fortune teller read my fortune for this year.
My fortune today was really bad.
I look forward to checking my horoscope's fortune every morning.
Apparently my fortune will improve starting next month.
運 is a casual, everyday word for luck in general, while 運勢 suggests a more systematic reading of one's fortune, as in fortune-telling.
運勢 refers to one's fortune as told through horoscopes or divination, whereas フォーチュン is a borrowed word rarely used in ordinary speech.
運勢 refers to one's fortune or luck at a given time (as in horoscopes), while 運命 refers to one's overall predetermined destiny.
Compound of 運 (luck, fate) and 勢 (force, energy), suggesting the momentum or trend of one's fortune.