G-d made the two great lights: the greater light to rule the day, and the lesser light to rule the night . . . And it was evening and it was morning, a fourth day. (Genesis 1:14, 19)
One who sees the sun at its turning point should say, "Blessed is He who reenacts the works of Creation." And when is this? Abaya said: every 28th year. (Berachot 59b, Babylonian Talmud)
Birkat Hachamah (BEER-kaht hah-chah-MAH) is a special blessing recited once every 28 years, commemorating the work of Creation and acknowledging G-d as the Creator of all things, even the sun that many ancient cultures worshipped.
The Torah teaches us that the sun was created on the fourth day (Tuesday night/Wednesday day), and tradition teaches that it was created at the tekufat Nissan (the beginning of Spring; another tradition says the world was created on Rosh Hashanah). For this reason, we remember the work of Creation by reciting this blessing upon seeing the sun fully risen on the morning after a tekufat Nissan that occurs at the beginning of the fourth day of the week (that is, at 6 PM on a Tuesday), which occurs only once every 28 years. The blessing is recited only if the sun (or at least its light) is visible. If it is completely overcast, the blessing is not recited.
Because this blessing is fundamentally related to the sun, its date is calculated using a solar calendar, rather than the usual Jewish lunisolar calendar. The Jewish solar calendar, like the ancient Julian calendar, assumes that the solar year is exactly 365¼ days. This calendar was established by the Talmudic sage Samuel and recorded in the Babylonian Talmud, Eruvin 56a. The rabbis of ancient times were well aware that this calculation was not entirely accurate -- indeed, Rav Ada in Talmudic times developed a somewhat more accurate calendar -- but the rabbis opted for simplicity over astronomical precision so the mitzvah could be understood and observed even by the mathematically challenged. In addition, if a truly accurate calendar were used, tekufat Nissan would never occur at precisely the beginning of the fourth day of the week, eliminating this observance altogether! Nevertheless, the discrepancies have already become quite significant over time. Birkat Hachamah was recited on March 25 of the Julian calendar (which was the equinox about 2400 years ago), but it is currently recited on April 8 of the more precise modern Gregorian calendar, about 18 days after the equinox.
A fundamental part of the solar calendar is the beginning of the seasons, which are referred to as tekufot (singular: tekufah; combined form: tekufat). Some say that the term would be properly translated as "turning points," but the word is usually translated as "equinox" when referring to tekufat Nissan (Spring) or tekufat Tishri (Fall) and as "solstice" when referring to tekufat Tammuz (Summer) or tekufat Tevet (Winter). These tekufot are used for timing Birkat Hachamah (which falls on tekufat Nissan), and also for timing an addition to prayer: in the Shemoneh Esrei prayer, the 9th blessing (Birkat Hashanim, the Blessing of the Year) inserts a request for rain at the beginning of the agricultural cycle, the 60th day of tekufat Tishri as calculated by Samuel.
Why does Birkat Hachamah occur only once every 28 years? It occurs only when tekufat Nissan occurs on the fourth day at the 0 hour (Tuesday at 6 PM). A solar year is 365 days and 6 hours, which is 52 weeks, 1 day and 6 hours, so any given date on the calendar advances one weekday and six hours every year. As you can see from the table on the right, tekufat Nissan occurs at the 0 hour of Day 4 every 28th year.
Because it is tied to the solar calendar, tekufat Nissan falls on the same day of the secular calendar every time it is observed (more or less). In the 20th and 21st centuries, tekufat Nissan April 8. However, because the Jewish solar calendar does not have the adjustments of the Gregorian calendar used by most of the world, the date of tekufat Nissan on the secular calendar advances when the year ending in 00 is not a leap year. In the 19th century, the tekufat Nissan was April 7; it changed to April 8 because 1900 was not a leap year; it remained April 8 in the 21st century because 2000 was a leap year; in the 22nd century, it will be April 9 because 2100 is not a leap year.
However, because the Jewish calendar is primarily lunar, this observance occurs on different days on the Jewish calendar: 4 Nissan 5741 (1981), 14 Nissan 5769 (2009), and 23 Nissan 5797 (2037), etc. It can occur anywhere from late Adar to late Nissan, and it is possible for Pesach (Passover) to start before tekufat Nissan (before it is considered to be Spring).
In 2009, Birkat Hachamah fell on Erev Pesach, the day before Passover. In other words, the blessing was recited on the morning immediately before seder night. This timing is very rare: according to an Orthodox Union article, it last occurred in 1925, but the last time it occurred before that was 1309! A Star-K article indicates that the last time this occurred before 1309 was 693 and 609.
A respected Chasidic rabbi, Rabbi Meir Yechiel HaLevi of Ostrovtza (known as the Ostrovster Rebbe), said that the Exodus from Egypt and the events behind Purim occurred after a Birkat Hachamah on Erev Pesach. The Rebbe suggested that the final redemption would occur after a Birkat Hachamah on Erev Pesach that was coming soon. Of course, he said this before the 1925 Birkat Hachamah on Erev Pesach, so most people assumed he was talking about that one, and that he was wrong. Some have also quibbled with his dating of prior redemptions, noting that the traditional dating of the Exodus (Jewish Year 2448) and the Purim story (around Jewish Year 3405) are not years when Birkat Hachamah is said. More forgiving sources suggest that those miraculous redemptions occurred within the 28 year cycle after such a Birkat Hachamah (although I don't know if the date calculations would work for that; dates weren't calculated mathematically then anyway), and that the cycle coming "soon" that he was talking about may have been 2009 rather than 1925. Certainly, 2009 was "soon" enough that many people who were living in 1925 were still living in 2009.