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?
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.
"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."
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".
I've gathered a bunch of sources, mentioned in the answers already here, and in other places.
While Rashi to Mishlei 23:2 says that a Bal Nefesh is:
אם רעבתן אתה ותאב לאכול
If you are glutton and desire to eat
[אם יש לך נפש משכלת למאוס ברע - If you have a discerning soul to hate evil]
When halacha refers to someone as a Bal Nefesh, they are referring to someone on a higher spiritual level.
In Talmud Pesachim 40A, Rashi translates a Bal Nefesh as a Chasid, a pious person. The Mosif Rashi on the Daf brings the Rashi from Chulin 6A (quoted below) and adds from "Ha'Orah Chelek 1 Siman 84 (anyone know what that is?)"
"someone who rules over his inclination and distances himself from sin"
בעל נפש כלומר המתרחק מריח עבירה ומדקדק הרבה על עצמו A bal Nefesh is one who distances himself from (even) a whiff of sin and is very exacting on himself.
Rashi on Talmud Chulin 6A translates a Bal Nefesh as an "Adam Kasher". Micropedia Talmudit brings two opinions about the definition of an Adam Kasher is (it appears they are both brought in the Tur and commentaries of Yorah Deah 346, but I did not look them up):
Rosh in the name of the Maharam -- Someone who is not suspected of any sin, nor of neglecting any positive commandment. Nothing sinful is ever associated with him. This applies even if he is not a great Torah scholar.
Bach, and others. -- He is recognized as being fit/proper, he seeks out Mitzvot and good deeds, even if he is not learned.
The Aruch brings an additional interpretation that I did not understand. I'll put it here, and if anyone can translate it and/or bring a source, please add it in:
וי"מ בעל נפש דכל שעה שאין לו אנינות הדעת שיכול לאכול כל פת
The Hon Ashir (by Immanuel Chai ben Ben Avraham Ricchi (https://www.jewishencyclopedia.com/articles/12736-ricchi-raphael-immanuel-hay-ben-abraham), the author of the Mishnat Chassidim) on Avot 1:17 says:
ואם בעל נפש אתה, שאי אתה חש על כבוד גופך כי אם בתורת ה' חפצך
If you are a Bal Nefesh, meaning you do not care about the honor of your body, rather you desire the Torah of G-d.
R' Chaim of Vholozin, in his Ruch Chaim to Avot chapter 3, end of mishna 1, says that a person is given free choice to transform his physical corporeality into spirituality, should he desire to do so. The opposite is also possible. Those who transform their flesh into spirit are called "Baalei Nefesh". [I recommend reading inside, because it is wonderfully written, and there are many great details I left out]
The Ohr Hachayim to Vayikra 22:12 explains that there are different levels to the soul, which each have different purposes. The Nefesh is the base level that all created beings have when they're born, but the Nefesh alone does not enable the person to elevate his physical actions and make them spiritual. That is only possible once a person receives his Ruach (second level) by virtue of his good deeds. Once he has a Ruach, he is able to elevate his Nefesh to the level of Ruach.
Thus, a Bal Nefesh is one who is able to elevate his Nefesh to the level of Ruach. (read the Ohr Hachayim inside for many more details).
The Ohr Hachayim on Vayikra 4:2 says that when a person sins he causes damage to his Nefesh (which is why a wicked person is called dead even while he lives). Thus, a Bal Nefesh is someone who hasn't sinned, even unwittingly. [He goes on to explain that the sin does not complete damage the Nefesh, and it unwitting sins can be expunged through sacrifices, and intentional sins can be expunged through Teshuva and Yom Kippur (read the Ohr Hachayim inside for many details).
(an interesting discussion from a Chabad perspective about whether a Bal Nefesh is a higher or lower level than a Yirat Shamayim here )