I have seen "Baal Nefesh" used frequently in various contexts: sometimes as one who is scrupulous in halacha, other times as someone who is a quality person.
What is its simple translation/meaning?
"Someone who really cares about their soul"; a "soul man", if you will. Often "the average Joe need not be stringent about this, but a soul man should be." If I'm not mistaken, Rabbi Moshe Feinstein occcasionally does rank stringencies, and I think he says "appropriate for anyone G-d fearing" is stronger than "appropriate for any soul man."
SBA quotes the Yismach Mosheh that a "ba'al nefesh" is someone for whom the things related to the soul are more important than physical things. Rabbi Gil Student also quotes some statements by Rashi and Rabbeinu Chanan'el to define a ba'al nefesh. Rabbi Micha Berger quotes the Nefesh HaChayim as well; see there.
You can read the whole subject on Avodah here.
Jastrow demonstrates (מר IV, p. 834) through a number of Talmudic sources that the term was popular during the Second Commonwealth and the meaning is, specially in this construct, "master over his desire".