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Yom Kippur is indeed the Jewish Day of Atonement. Its formal name yom ha-kippurim means "day of/for atonements." It is an ancient holiday, being one of the festivals decreed in the Torah (see chapters 16 and 23 of Leviticus).

The goat reference is to the Biblical celebration of Yom Kippur that they did in the Mishkan (tabernacle) and the Temple, thousands of years ago, when Israelite Jews offered sacrifices. At that time, the kohanim (priests) in the Temple would help people expiate sin by means of sacrificing sin-offerings, generally goats. This was done by the priest placing his hand on the goat, ritually transferring the sin in question onto the goat, and then ritually slaughtering the goat and burning its body on the altar. On Yom Kippur, in those days, the High Priest did this on behalf of all the people, but used two goats: one was confessed upon as usual and offered up, but the other was sent into the wilderness, there to be killed by a kind of stoning. In Hebrew this goat was called the se'ir ha-mishtalei'ach, or "the goat that is sent forth;" the King James, in translating this, uses the term 'scape-goat, 'scape being an archaic shortening of "escape," which in Tudor-era English meant "to run forth," (not necessarily "to get away from"). However, all of this is reflective of a Yom Kippur tradition that has not been practiced in close to two thousand years. We have done things very differently for a long time now.

In any case, together with Rosh Hashanah (the Jewish New Year), which comes ten days before it, Yom Kippur makes up what are called in English the High Holidays, or in Hebrew, yamim ha-nora'im, The Days of Awe.

Judaism is, by and large, not a religion that spends an excessive amount of time concerning itself with formal confessions and absolutions of sin. For the most part, one's sins are something that one is expected to deal with on a daily basis, by avoiding sinning if possible, and if not, by doing teshuvah, which is loosely translated as "repentance," although it has a deeper meaning, something akin to "turning oneself around [away from sinful behavior, and going back to positive behavior]."

(Just for your reference, nearly all common sins in Judaism are forgivable. If one sins against one's fellow person, one must ask for forgiveness, pledge to avoid such behavior again, and make any recompense or restitution needed. If one makes such pledges and recompense, and asks three times for forgiveness without receiving it, he is held forgiven, and the other party is considered to have transgressed. If one sins against God-- say, by transgressing a ritual commandment, like eating something non-kosher-- one pledges to try and refrain from such behavior in the future, and one asks forgiveness from God, who, we generally suppose, grants it.)

But we have a process of formal introspection, confrontation of one's sins and faults, and a communal time and place to ask God, as a people, for forgiveness for all our sins for the year. This process takes place beginning on the first of the month of Elul (this year 31 August: remember that all Jewish holidays run on our Hebrew calendar, not the Gregorian calendar, which is Christian. And because we use a lunar calendar, it doesn't always match up with the Gregorian solar calendar in the same ways, so Jewish holidays are sometimes said by Jews accustomed to living on the Gregorian calendar to "come late" or "come early" in a given year, although of course by our reckoning, they always fall on the same days.), where one begins introspecting, and seeking forgiveness from anyone that one suspects that one may have transgressed against in the previous year, and praying meditations to remind ourselves that Yom Kippur is coming, and that we must pay close attention to our behavior, and what needs we have to pledge ourselves to self-improvement in the year to come (which will begin with Rosh Hashanah, the first of the month of Tishrei).

The services for Rosh Hashanah are special, unlike the services for other holidays the rest of the year, and all are dedicated to commanding the attention of the people to prepare themselves for a spiritual reckoning, and to be ready to confront their own personal transgressions, and those we all communally take responsibility for. This holiday begins a ten-day period concluding with Yom Kippur which are called the aseret yemei teshuvah, or Ten Days of Repentance, wherein whatever preparation one has not done during the month of Elul is completed, so that one is ready for Yom Kippur.

Yom Kippur begins (as all days in Jewish reckoning do) at sundown of the day previous. Prior to sundown, a last meal is taken by families together, for at sundown, the fast begins. From sunset to sunset, Jews do not eat, drink, bathe (any more than is minimally necessary for hygiene), anoint themselves with perfume or cologne, overly adorn themselves with excessive makeup or jewelry, or wear any products made of leather. At sundown, we go the synagogue, and there is a special evening service, wherein we begin the first of numerous repetitions of formal, communal confessions of the sins of the People Israel, and whereat we annul in advance any ritual vows we may make during the coming year (this is not applicable to the everyday sort of promises and contractual obligations that one encounters in daily life, but only to religious vows made to God), thereby sparing ourselves consequences should we inadvertently make such vows, or, having made them, fail to fulfill them, or break them.

We return to the synagogue the next morning, and a very long service begins. Jewish services tend to be longer than Christian or Muslim services anyhow, but Yom Kippur services are long even for us. Most begin around 9 am, and run until 2pm or so. There are many communal confessions, and the liturgy is devoted to recalling God's justice, and His right as our Maker to judge us, and His endless mercy in forgiving us, time and again. We recall the merits of our ancestors, and the many trials we have undergone in trying to remain a faithful people to God, asking Him to take those into consideration when judging us now. And we recall the Temple service, reciting an account of it in place of actually doing it. Yom Kippur is basically the only time of the year, also, when Jews prostrate themselves in prayer, which we do several times on that day.

Then there is usually a short break: some study Torah at this time, others go home and nap. After an hour or two, we return to the synagogue for the afternoon service, following which is a special evening service, where we note the coming of the day's ending, and send our last prayers to God for forgiveness and a clean slate for the coming year. When this is over, a blast is blown on the shofar (a ram's horn, blown often on Rosh Hashanah), and the day is done. At this time, people usually go and have a celebratory fast-break meal, with friends or family.

Every year.

Most of the traditions involve what is done at the services in terms of alterations and additions to the liturgy. Special liturgical poems, only read on Yom Kippur, are added to the services, special melodies are used; there is even a special cantillation for the High Holidays, a different set of chanted notes by which portions of the Torah are read aloud to the congregation. There is the fasting, of course, but that is a law, not so much a custom. Many people wear white on Yom Kippur, as a sign of purification; some wear a white robe called a kittel, which symbolizes the shroud worn by the dead, as a reminder to us that we are mortal, and our Creator has the right to be our judge. Some have the custom of, prior to the beginning of Yom Kippur, lighting memorial candles to burn through the day, recalling family members we have lost. And there are certain differing prayer customs, also; such as some having the tradition to stand for most of the evening and morning services.

Yom Kippur is not really a time when extended families come together, unless they happen to live in the same community. It's not really a celebratory holiday, but solemn, and, as you can see, mostly observed in the synagogue, so it's not very conducive to family gatherings. Some extended families do come together for Rosh Hashanah, though. And some come together for Sukkot (the week-long Feast of Shelters), which comes five days after Yom Kippur. But in my experience, if there's a Jewish holiday that many extended families come together for, it's Passover.

It does. Maybe even more so, as the theology has only deepened and become ever more complex and nuanced over time.

Very, although mostly that has to do with there no longer being sacrifices, as I mentioned earlier, and with the development of our liturgical tradition, which really began in a formalized way after the destruction of the Temple (70 CE).

Yom Kippur observance is actually pretty similar whether in Orthodox or non-Orthodox communities. The big differences are the same ones that one finds at any time in Orthodox vs. non-Orthodox liturgical services and synagogue rituals: Orthodox synagogues pray exclusively in Hebrew, and use the fullest version of the traditional liturgy, without additions or subtractions of any note; men and women are seated separately and have a barrier between them; there is no instrumental music, and most often no choral music; and people are generally stricter and more rigorous in their interpretations of the laws and customs. Whereas in non-Orthodox synagogues, most communities modify the traditional liturgy to some degree, even if only to add a few prayers in English (though most Reform congregations modify the liturgy extremely, substituting considerable English and creative liturgical poetry, geared toward a community that is much less Jewishly educated). Some non-Orthodox synagogues employ organs or pianos, or even guitars, violins, or cellos. Some employ choirs. Men and women are seated together. And there is more leniency surrounding the customs of the day, and a certain amount of flexibility regarding certain details of the laws.

There's always some minor variation, but for the most part, the observance of Yom Kippur, probably more than nearly any other holiday, is fairly uniform. If there are major variations in custom, they are due more to difference in denomination than in location, I would think.

Yom Kippur always falls on the tenth of the month of Tishrei. On the Gregorian calendar, that can be as early as the beginning of September, or as late as mid-October. But it falls on the tenth of Tishrei because that is when the Torah says it must be observed.

I hope that covered the basics. Please feel free to ask about anything I didn't clarify, or anything else you might wish to know. I am very delighted to have the chance to share knowledge.

__________________
Dull sublunary lovers love,
Whose soul is sense, cannot admit
Absence, because it doth remove
That thing which elemented it.

(From "A Valediction: Forbidding Mourning" by John Donne)