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".
Most literally the phrase Baal Nefesh is probably something like "master of [a] throat", although the simple meaning is more like "possessor of a soul". Either way though it seems to include just about everyone.