tl;dr: This year, on the east coast of the US, the autumnal equinox occurs during Yom Kippur, a coincidence that is rare and that depends on your location. But 2015 is even more rare and special in that the equinox falls on Yom Kippur almost worldwide.
(Apologies if all this has already been discussed.)
In 2015, Yom Kippur---the Jewish Day of Atonement---begins at sunset on September 22 and ends at sunset the next day. The autumnal equinox occurs at 4:20 AM ET on September 23. So this year, on the east coast, Fall begins during Yom Kippur.
When did this last happen, and when will it happen next? The answer is a little ambiguous. In 2034 the equinox occurs at 6:38 PM ET just as the holiday is starting. This means that whether autumn begins during Yom Kippur depends on the local time of sunset, which varies within the timezone, or on when services start at your synagogue. Similarly, in 1947 the equinox took place at 5:29 PM ET erev Yom Kippur, probably too soon, and in 1939 it was at 6:49 PM ET that evening, which was more likely during the holiday.
To find a year with no such ambiguity we must go back to 1901; the equinox took place at 2:09 PM ET on Yom Kippur day. Or we can wait until 2053 when it will occur at 9:04 AM ET during the fast.
The discussion above applies only to the east coast. More below.
The start and end of Yom Kippur are defined by local sunset but the equinox is at the same instant worldwide. To understand the implications, look again at the ambiguous 2034, where the equinox is at 6:38 PM Boston time erev Yom Kippur. That's 3:38 PM Pacific Time, so in California the equinox will definitely precede the holiday. And it's 12:38 AM in London and thus is certainly during Yom Kippur. As another example, in 2053 the equinox is at 10:04 PM in Japan and, unlike in Boston, will occur after it's all over but the eating.
Is it possible that in some year Fall will begin on Yom Kippur everywhere in the world? I don't think so. The closest we can get is when the equinox occurs just as the holiday ends in the part of the world with the latest time---that is, just west of the International Date Line. Sunset there is dawn UTC, roughly 3:00 AM EDT, which is close to what happens this year! So 2015 is extremely special in that the autumnal equinox occurs during Yom Kippur very nearly across the planet, missing perhaps a few thousand Jews in New Zealand but not many more. It doesn't get better than this until Thursday September 22 2664, when the equinox is at 3:59 AM EDT the morning of Yom Kippur. Bring your shofar.
I'd be happy to hear a more definitive answer to these questions. We need someone with persistence and nerditude to track the problem through several complications, e.g. the fact that the Date Line itself moves a couple timezones each direction, the length of the fast (officially 25 hours), and the actual times you should celebrate Yom Kippur if you're near the IDL or the poles. (There is still indecision over the location of the "Jewish IDL": When it's noon Saturday in Jerusalem, where is it no longer or not yet shabbat?)
Technical notes: I'm taking the autumnal equinox as defining the start of autumn; this convention is not universal. I'm also pretending that the rules for Daylight Savings Time have always been and will always be the same as they are today, a sort of "proleptic" DST. For example, there was no DST in 1939, so the equinox was actually at 5:49 PM, not 6:49 PM. This is completely irrelevant to the conclusions.