[The original title of this piece was “Torah and Karma.” However, I found that people who wanted information about this were using the search words “Judaism” and “Karma.” So, I changed the title as above, to make it easier to access this post].
Once, around 1980, attending a class on Hasidut, I heard a participant ask the instructor, who was a Lubavitcher Hasid, “Do Jews believe in karma?”
The instructor, who was highly well-versed in Hasidic teachings (especially HaBaD), said that he wasn’t familiar with what “karma” meant. [6/12/14 note: This would be far less true today than it was then.]
In Sanskrit, “karma” literally means “action” but is more generally understood as the automatic, proportionate reaction to our actions. A “good” action on our part will produce “good karma” — a good effect on or for us; a “bad” action will produce “bad karma” — a negative effect on or for us.
“Any work, any action, any thought that produces an effect is called a Karma. Thus the law of Karma means the law of causation, of inevitable cause and [effect]. Wheresoever there is a cause, there an effect must be produced…[the law of Karma] is true throughout the whole universe.” 
It’s a fundamental teaching in yoga and other Indian traditions. Those better-versed in Buddhism than I am would probably agree that it’s fundamental in all branches of that tradition, too.
So, is it a “Jewish” concept?
I’d say: Yes, but not when it’s expressed as an impersonal “law” — like “gravity” — in which G-d isn’t mentioned or has no part.
More typically, Torah speaks of the reactions to our actions as coming from the Divine:
במדה שאדם מודד בה מודדין לו
“With the measure one measures, They measure out to him [or her].” 
“They” here means “The Heavenly Court” — the Divine setting in which G-d is described as deliberating and judging, after having weighed all possible arguments, pro and con. But this can be understood as figurative, as we’re consistently also taught that “the whole world [or: the entire Creation] is filled with G-d’s ‘glory’.”  G-d’s Presence is always everywhere. So, the responses to our actions are coming from the Divine — a Presence in this world as well as in Heaven — and aren’t simply neutral; they’re always for the Good.
Yet, in both traditions, the response to our actions is “spontaneous,” even “automatic,” the description of G-d “deliberating” notwithstanding. G-d might be said to “moderate” a response, according to specific circumstances, but never to suspend it arbitrarily. In this sense, “karma” can be to at least some degree congruent with Jewish belief.
Another similarity with “karma” is that “Their” reaction is in proportion to what a person has done. (That Divine “Justice” can be tempered with “Mercy” should be the topic of a future discussion about this, too.)
“Karma” also impressed on me, when I was first exposed to the concept, that the effects of our actions are utterly inescapable. An “inevitable” response is certainly Torah, too:
הוא ראה גולגלת אחת שצפה על פני על המים אמר על דאטפת אטפוך וסוף מטיפיך יטופון
“[Hillel] saw a skull floating on the water. He said, ‘Because you drowned others, they [*] drowned you; in the end, those who drowned you will themselves be drowned’.” 
So, even if we say that an “impersonal force” or “law” of karma isn’t a “Jewish” teaching, it’s undeniable that Torah [even in the wider sense of traditional Jewish teaching] tells us that there’s an inevitable, proportionate, “karma-like” response to our actions — from the Divine.
The teaching is fundamental to Judaism: 
“Hillel was teaching the world the important lesson that G-d’s Providence [השגחה] is specific [פרטית]. [G-d] constantly views and evaluates each individual’s needs and recompenses each one measure for measure [מדה כנגד מדה]. Nothing is without reason.” 
In the 18th century CE, The Baal Shem Tov (the Besht), founder of the Hasidic movement, taught the same thing in his “earthy” way:
“…When a living creature stands by its shadow…
just as the creature moves, so the shadow moves.
It is quite the same with [G-d].
According to the actions of a mortal human being on earth,
so he is treated by the Heavenly realm above.” 
So, from the Jewish point of view, “karma” doesn’t exist as an abstract law separate from G-d, but as a categorical statement of the way that the Divine characteristically responds.
Knowing that the effects of even our least significant actions are utterly inescapable should give us considerable hesitation before acting thoughtlessly. Such “hesitation” is called the “fear of G-d.” It means recognizing the impossibility of avoiding the effects of what we do.
Yet, Judaism looks on that “fear” positively, as the true beginning of our spiritual development:
“ראשית חכמה יראת ה
“The beginning of wisdom is the fear of G-d” 
“That feeling must be the starting-point of a life which is to be lived on a higher plane than animal existence.” 
That’s why many people today prefer to translate “fear” as “awe.” We need not misunderstand “fear” as requiring pointless guilt or useless anxiety. Instead, it’s meant to stir us to choose actions (and thoughts and words) that are best, from the widest possible range of consideration.
All comes from G-d, as Rabbi Levi Yitzhak of Berditchev sung in “A Dudele” :
Something’s good — you. !איז עמיצן גוט – דו
Bad — oy, you. !שלעכט – איי,דו
Recognizing that all comes from G-d — always; even the “bad stuff” — can also bring us into a feeling of G-d’s intimate closeness to us. That closeness can bring us the Comfort and Peace of G-d’s own Presence and Goodness.
Then, we can find G-d as our surest Comfort, even in the hardest difficulties:
“G-d is my Rock…” 
Who wouldn’t want to live in harmony with a Peace that’s always there for us?
And — as someone said (I can’t provide a citation for this), “Fear…is ‘the beginning of wisdom’. But only the beginning.”
We should never forget, though, that whether we regard suffering as “karma” or Divine Judgment, Torah clearly teaches us that we have an eternal responsibility to ease the suffering we see around us. We might accept that the suffering ultimately has a Divine Source for the good, but the Divine also expects us to care for and to comfort those who are suffering — even if as the result of their own actions.
The final goal of wisdom is Love.
(for further discussion of Judaism and Karma, see:
 Swami Vivekananda; Karma Yoga; p. 97
 Mishnah Sotah 1:7; p. 8b
 Yishiyahu/Isa. 6.3
 Pirkei Avot 2:6 (Judaica Press edition; 2:7 in other editions)
[* In the present usage, “they” refers not to the Heavenly Court, but to the people who have been “appointed,” as it were, by that Court to carry out the execution. The Talmud discusses this question elsewhere.]
 Some contemporary branches of Judaism — e.g. Reform; Humanistic — might question or deny a belief in “reward and punishment,” but I’d let my statement stand as is, because the concept is still one that Reform and Humanistic theologians recognize as one that must be addressed. Their recognition is based on its prominence in older, more traditional Jewish teachings. If I’m mis-representing Reform or Humanistic theology, correcting comments are most welcome.
 Rabbi Yitzchak ben Mosheh Magriso; Ethics of the Talmud (Me’am Lo’ez), Pirkey Avoth (1747); translated from the Ladino by David N. Barocas; edited by Rabbi Aryeh Kaplan; Maznaim Publishing Corporation, c. 1979; p. 85
 Dvorkes, Rabbi Aryeh and Joshua; Chas. Wengrow, trans.; The Baal Shem Tov on Pirkey Avot; © 1974 by Rabbi Y.A. Dvorkes; p. 23 (from Rabbi Levi Yitzhak; Kedushat Ha-Levi); p. 23;
The Besht is commenting on Ps. 121:5 — G-d is your shadow/ ה׳׳ צלך (usually translated “G-d is your shade…” meaning that G-d is your comfort or relief)
 Ps. 111:10; see also Pr. 1:7
 Soncino Press commentary on The Psalms; p. 375
 Ps. 18:3