All three words can mean something like "soul", e.g., neshama in גלגול נשמות, and ruach in וַתְּחִי רוּחַ יַעֲקֹב אֲבִיהֶם (Bereshit 45:27). So, what is the difference?
The names of nefesh, ruach, neshamah, yechidah, chayah (in that order) is from Bereishis Rabba 14:11.
I wish to add two more positions to the discussion.
The first is Rav Saadia Gaon's (Emunos veDeios 6:3). Rav Saadia Gaon was an Aristotilian rationalist, although to a lesser extent than the Rambam, who had to deal with the topic when writing a commentary to Seifer haYetzirah:
.. When [the soul] is attached to the body, one can see it three abilities: - the power of choice - the power of desire - the power of anger Therefore it is called in our language by three names: nefesh, ruach and neshamah.
It is hinted in the name "nefesh" that is has the power of desire, "for the taavah of your nefesh" (Devarim 12:20), "his nefesh was satisfied of desire" (Iyov 33:20).
And it is hinted in the name "ruach" that is has the power of being annoyed and angry, when it says "do not become empty through your ru'ach by getting angry" (Qoheles 7:9), "his entire ruach brings out foolishness" (Mishlei 29:11).
And it is hinted in the name "neshamah" that it has the power of wisdom, as it says "the 'Neshamah' of Shakaai will understand them" (Iyov 32:8), "and the neshama of who came out of you" (Iyov 26:4)
Because of these [seprable] abilities, the one who made it to be two parts erred, that one of them is in the heart and the other in the rest of the body. Rather, all three are in the soul (nefesh) alone. So the language added two more terms, they are "chayah" and "yechidah". It is called "chayah" because it exists through that which its Creator persists it. But it is "yechidah" because there is nothing like it among all the [other] creations, neither in the heaven, nor on earth.
According to Rav Saadia Gaon, there is one indivisible soul, but it has three sets of abilities -- the emotions that draw us to people and things (desire), those that drive us away from them (anger), and thought. The soul has different names, depending upon context, upon whether we are talking about one of these three abilities, its ability to persist, or its uniqueness
The other opinion I wish to add or perhaps two opinions, is that of the Vilna Gaon and R' Chaim Volozhiner:
R. Chaim Volozhiner explains Nara”n based on the breathing imagery used in Bereishis 2:5:
וַיִּיצֶר֩ ה אֱ-לֹקים אֶת־הָֽאָדָ֗ם עָפָר֙ מִן־הָ֣אֲדָמָ֔ה וַיִּפַּ֥ח בְּאַפָּ֖יו נִשְׁמַ֣ת חַיִּ֑ים וַיְהִ֥י הָֽאָדָ֖ם לְנֶ֥פֶשׁ חַיָּֽה׃ And Hashem G-d formed man of dirt from the ground and He breathed into his nostrils a living neshamah; and the man became a living nefesh.
Rav Chaim writes (Nefesh haChaim 1:15):
Our Rabbis za”l already compared the three-fold living ru’ach of man to the making of a glass utensil to reviving the dead. They said, “It is a qal vachomer (a fortiori) argument from a glass utensil, which is made by the breath of flesh and blood… Flesh and blood, which is made by the breath of HaQadosh Baruch Hu, how much more so!”… For the message must be similar to the metaphor. When we study the breath of the mouth of the worker into the glass container when he makes it, we find in it three concepts. The first idea is when the breath of air is still in his mouth, before it goes into the opening of the hollow tube, we can only call it then a “neshimah”. The second idea, when the breath enters the tube, and continues like a line, then it is called “ru’ach” (wind). The third, lowest, idea, is when the breath goes from the tube and into the glass, and inflates in it until it becomes a container to fit the will of the glass-blower, then his wind stops and is called “nefesh”, a term of rest and relaxation.
The soul is likened to the breath of air that a glass blower uses to inflate hot glass. The Nefesh haChaim, following a far older metaphor, breaks down both processes into three parts. The first is the air, as it is still in the glassblower’s cheeks. This corresponds to the neshamah, the part of man which is most connected to Hashem. When the air leaves the glassblower’s mouth, it flows down a tube. The tube connects the glassblower and his work. In the same sense, the ru’ach dwells in the connection between the physical and the spiritual. This flow, a wind, is the ru’ach. From the tube, the air enters the glass, “dust of the ground”. This is the nefesh, giving shape and purpose to our physical selves.
This metaphor gives us another description of how the ru’ach, by being the decisor, also becomes a source of desires. Recall that I started this series with the notion that the ru’ach, man’s existence in the world of his own mind and in a relationship to himself, was a the person between the angel and the little devil propped on his shoulders. And yet now we’re saying it has desires too! The ru’ach is the connection between the nefesh and the neshamah, it exists because of the tension between being both Divine “breath” and clothed in earth. Yet, because it sits in this middle world, the ru’ach is also an entity in its own right.
The the Vilna Ga’on is in his discussion (Peirush al Kamah Agados, Koenigsburg ed. pg 10b) of the following gemara:
[The elders of the School of Athens said to Rabbi Yehoshua ben Chanania,] “Build us a house in the air of the world.” He pronounced a [Divine] Name, and [thereby] suspended himself between heaven and earth and said to them, “Send me bricks and cement from down there.” They said, “Can anyone find the ability to lift them to there?” [R. Yehushua] said, “Can anyone find the ability to build a house in the air?”
The Gaon explains that the house in the air is clearly a reference to the ru’ach, suspended between heaven and earth. (After all, the word “ru’ach” also means “wind”.)
The ru’ach has the ability to decide, and thus the concept of Free Will. With will comes a desire to see that will implemented, to make the worlds outside ones head match the world as we imagine it could be. With will comes a hunger for power and control. Rather than being the means to get things done, they can take over and satisfying them can become an end in itself.
A person has control over an object when he possesses it. And money gives a person more opportunities to get more of his dreams accomplished. When, sadly, someone turns it into an end in itself, they can never be satisfied. The hunger is for a means, which can only be put to trying to get more. “He who has 100 zuz, wants 200.”
This is an aspect of the nefesh as a whole. It’s clear that the role of mitzvos between man and himself are not given the same central role in Jewish discourse as those between man and G-d and between man and other people. This is because they are not an end. The point is not self-contemplation. To be the perfect self is to be perfect in one’s relationships, the same three relationships.
… including the relationship with oneself. This self-reference is a concept that comes up often when dealing with the concept of intelligence. And self-awareness, consciousness of one’s own thought, is the essence of the nefesh, of making Free Willed choices.
Accoding to the Rambam in "the Guide of the Perplexed", (and WAF above), the meaning of the words in each individual case are to be determined by the context."
Ruach is a homonym, signifying" air," that is, one of the four elements.
"And the air of God moved (Gen. i. 2).
It denotes also," wind." In this sense the word occurs frequently.
"And the east wind (ruach) brought the locusts" (Exod. x. 13):" west wind" (ruach) (ib. 19).
Next, it signifies" breath."
"A breath (ruach) that passeth away, and does not come again" (PS. lxxviii. 39) wherein is the breath (ruach) of life" (Gen. vii. 15).
It signifies also that which remains of man after his death, and is not subject to destruction.
"And the spirit (ruach) shall return unto God who gave it" (Eccles. xii. 7).
The Hebrew ruah when used in reference to God, has generally the fifth signification: sometimes, however, as explained above, the last signification, viz., "will."
The Hebrew nefesh (soul) is a homonymous noun, signifying the vitality which is common to all living, sentient beings.
"wherein there is a living soul" (nefesh) (Gen. i. 30).
It denotes also blood," as in
"Thou shalt not eat the blood (nefesh) with the meat" (Deut. xii. 23).
Another signification of the term is "reason", that is, the distinguishing characteristic of man, as in
"As the Lord liveth that made us this soul" (Jer. xxxviii. 16).
It denotes also the part of man that remains after his death (nefesh, soul)
"But the soul (nefesh) of my lord shall be bound in the bundle of life" (I Sam. XXV. 29).
Lastly, it denotes" will";
"To bind his princes at his will" (be-nafsho) (PS. CV. 22): "Thou wilt not deliver me unto the will (be-nefesh) of my enemies" (Ps. xli. 3):
and according to my opinion, it has this meaning also in the following passages,
"If it be your will (nafshekem) that I should bury my dead" (Gen. xxiii. 8):" Though Moses and Samuel stood before me, yet my will (nafshi) could not be toward this people" (Jer. xv. 1), that is, I had no pleasure in them, I did not wish to preserve them.
When nefesh is used in reference to God, it has the meaning "will," as we have already explained with reference to the passage,
"That shall do according to that which is in my will (bi-lebabi) and in mine intention (be-nafsht)" (I Sam. ii. 35).
According to Jewish Magic and Superstition: A Study in Folk Religion (admittedly, not the best source.)
The Neshama goes up to Hashem (olam ha-ba and techiat hamatim), and both the ruach and nefesh remain on earth. The Ruach stays with the body and never leaves it. The week after death, the nefesh wanters between the deceased's' home and grave, then infrequently comes returns to the grave until it is weaned from the grave by the end of a year.
In this Youtube video (5:49 seconds in), Rabbi Chaim Miller explains that:
Nefesh == Instincts of the Soul
Ruach == Emotions of the Soul
Neshamah == Intellect of the Soul
He does not bring his source.
In Kabbalah or Chasidut Nefesh means a bodily soul, or instinct, or inate. (All created things have nefesh). Ruach is often used in the context of "Ruach Hakodesh", which is a sort of divine inspiration, or an aspect of godliness /correctness of thought (morality is one example of that). Ruach can also be the basic life force of any living thing. Neshama is the higher level soul which is connected directly to Gd, it exists only in humans.
Outside of Kabbalah, Nefesh is the life force of any living being. Ruach is the animator of life, its what allows things to move, and Neshema is the combination of Ruach and Nefesh.
I've seen it explained as follows (in Wisdom, understanding, and knowledge: basic concepts of hasidic thought):
Nefesh - Every creating thing in the world has a Nefesh, the G-dly force that gives it life. Inanimate objects (such as rocks) have only this. (see here)
Ruach - Every living thing has a Ruach as well, the life-giving force that animates the living thing. (see here)
Neshamah - Referred to in Tanya as "an actual part of G-d". This is only found in the Jew. (see here)
Note: I've linked to the snippets view of google books searches. This is not ideal. If anyone knows of a better online source that says the same thing, please let me know.
The Maharsha in his Chidushe Aggadot (Berachot 3A, d"h "Mishmorah Rishonah Chamor") says almost the exact same thing. He says that there are 3 parts of the soul of man which are called "The Natural", "The Life", and "The Intellect". He says that Kabbalists call them the "Nefesh", "Ruach", and "Neshamah". See there for a little more elaboration.
Nefesh can mean either "breath" or something like "life force". "Ruach" is usually used to mean "spirit" in the sense of "enthusiasm" or "vigor". Not sure about neshema and how it differs from nefesh.
EDIT: Wikipedia has a concise summary of the kabbalistic usage:
Kabbalah separates the soul into three elements: the nephesh is related to instinct, the ruach is related to morality, and the neshamah is related to intellect and the awareness of God.
Nefesh is the brain, so it can die. Thinking animals have a nefesh. However, ruach is immaterial; not physical. Ruach is your holographic-like personality software that develops in your nefesh hardware brain. Animals do not have a ruach with higher level thinking and consciousness. As software can be transferred from one computer through a network to another computer, when you physically die, your ruach will be transferred from your hardware brain through the logos (universal neural network) to either Heaven (New Jerusalem) or Hell (the grave Place of Torment, and later the Lake of Holy Spirit Fire). People have often wondered how the nefesh and ruach can be different if they both think and feel emotion. Now you know, it's because the ruach is the personality software that develops in the nefesh hardware.