The Hebrew calendar is a combination of solar and lunar measurements, and this enables to set the timing of religious festivals as indicated :
And God said: "Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days and years." (source: Genesis 1:14)
So, in a sense, the Hebrew calendar follows the directive of the Bible to use both luminaries as signs for the timing of seasons etc. In comparison, the secular calendar, which follows the Christian calendar, is only based on a solar year of about 365.25 days. And the Muslim calendar is only based on the lunar month of about 29.5 days. The use of one luminary only (the sun or the moon) necessarily leads to issues. On one hand, the solar calendar is correct with seasons, but it is incorrect with days in the month. As of the lunar calendar, it is correct with days and months, but annual festivals always fall at various times of the solar years / seasons.
The Hebrew calendar is more complex as it counts the days and months based on the lunar month (about 29.53 days), so each Hebrew month has 29 or 30 days depending on the observation of the moon, and it counts the years based on solar years (about 365.25 days) and restarts this count with the return of Spring. It means that, in some years, the calendar needs to add an entire extra month, so that the adjustment to Springtime would coincide with the actual season (in the Land of Israel). This extra 13th month is added to the last month, being the Hebrew month of Adar, and is simply called "Second Adar". It generally falls in February / March in the secular calendar. In Athens of Ancient Greece, they also used a lunisolar calendar, and added a 13th month occasionally to align their calendar to the seasons.
It is worth noting that the Hebrew calendar is also based on a cycle of 19 years: all the planets return to whatever position as they were 19 years earlier. This cycle was known in Ancient times as the Metonic cycle (because History accredited its discovery to the Greek Meton in the 5th century BCE). This cycle has exactly 235 lunar months, which means 235 * 29.53 days = 6939.55 days which equal to about 19 solar years ; indeed 19 * 365.25 = 6939.75 days. In fact, the Metonic cycle has either 6939, 6940 or 6941 days. The years when a second month of Adar is added are the following ones out of the 19 years cycle: years 0, 3, 6, 8, 11, 14, 17.
The Metonic cycle means that, if the first of a lunar month would fall on -say- the fifth day of a week in any given year, the first of the same lunar month would also fall on a fifth day in the year that will be 235 lunar months later (or 19 years later). So, for example, as the Bible states that Adam and Eve were created on a Friday (it was the 6th day of Creation, just before God established the Shabbat on the 7th day), it means that every series of 235 lunar months (or 19 years), from that specific day, also falls on a Friday.
Historically, the Jews of the times of the Temple used to have defined cities / locations that would watch the signal initiated from the Temple to indicate the start of a New Moon, and thus the start of a new lunar month. However, due to persecutions and exiles, it was no longer possible to rely on visual observation and therefore, in about year 359 CE, and due to initial persecutions and restrictions against Jews in the start of the Byzantine era in the ancestral land of Israel, the head of the Sanhedrin, Hillel the Second, decided to set a Hebrew calendar based on astronomical computations, and no longer on observations. This came at a time of great tensions between Jews and Christians over the question of the date of Easter, which set the time of Jesus' crucifixion. The issue rose at the Council of Nicaea (325 CE). Because Jesus is believed to have been crucified on a Friday, before the festival of Jewish Passover (Pesach), that day would necessarily be the 13th or 14th of the Hebrew month of Nisan. With a crucifixion on a Friday, the following Sunday became the day of resurrection (because the Gospels mentions Jesus resurrected "on the 3rd day"). However, in the Hebrew calendar, the 13th or 14th of Nisan do not always fall on a Friday: it depends on the discrepancy with the solar year. But, since the Christians wanted these Easter festivals to always fall on a Friday / Sunday within their adopted (solar) calendar, this did not match the Hebrew festival of Pesach which was the time of Jesus' crucifixion. Hence the dispute of calendar.
There are a lot of books and Internet sites explaining the Hebrew calendar in more details, so the readers can look at them to get more in-depth knowledge on the matter, if wished. Here I will limit myself to talk about the last month of the Hebrew calendar, a month called Adar or the "12th month". It is worth noting that the "last" month does not mean that the next month would start a "new year". This is because, in Christian/secular calendar, there is a connection between months and years (or new year), but there is no such connection in the Hebrew calendar, so the years are counted independently from the months, and the matching is done through the 19-years cycle with special years when a second month of Adar is added. In fact, the Jewish "New Year" falls at the beginning of the 7th Hebrew month called Tishri. Or let's put it this way : the Hebrew calendar has several (in fact four) "new years" depending on if you look at seasons, months cycles, agriculture, etc.
The Hebrew month of Adar is linked to many events of Jewish history. Above all, it is the month of the festival of Purim, related to the Book of Esther in the Bible.
In these times, an evil minister called Aman (or Haman) wanted to eradicate and kill all the Jews of the Persian empire. He consulted the astrologers to determine what month would be best to execute his plans successfully and they told him to use the month of Adar. But his plans failed, and Adar was changed from a month of potential grief to a month of joy. Since these times, the festival of Purim was added to the Jewish year of festivals and celebrations. There is a Jewish saying: "when Adar comes, Joy is increased". In term of etymology, the word Purim means "lots", because Aman had casted "lots" to decide when to exterminate the Jews. But, ultimately, the odds turned against him.
The festival of Purim is celebrated on the 14th and 15th of Adar, which, as it is the middle of a lunar month, always falls on a Full Moon (while the beginning of a lunar month is always a New Moon). Adar is celebrated on the 14th in any normal city, but it is also celebrated the following day, on the 15th, in "walled cities" (today, only in Jerusalem). Jews all over the world are supposed to do four things on the festival day: read the Book of Esther, get a festive spirit, rejoice, and offer gifts to other people especially to the needy and the poor. This festival of Purim is so important that, the Jewish Sages stated : when the Messiah will come, all festivals will be ceased except for...Purim. (source: Yalkut Mishlei 944).
The month of Adar also witnessed other events of Jewish history and in the Bible.
3 Adar: the Second Temple was completed (source: Ezra 6:15)
7 Adar: date of birth and of death of Moses, who lived 120 years; it is worthy to note that the 7th of Adar never ever falls on a Shabbat
12 Adar: God exacted vengeance against "Turyanus" (probably a nickname derived from tyrannus which means "despote" in Latin; the word comes from the Greek τυ ́ραννος which means "absolute master"); he was a Roman officer and wanted to kill the Jews of Lod/Lydda
13 Adar: the Fast of Esther because she, herself, fasted before her dinner in presence of her husband, the Persian king, and his minister Aman. The story is related in chapter Esther 9. This Persian king, called Ahasuerus in the Bible, was Xerxes who had just returned from his ill-fated campaign against Greece and his two defeats, one on land in the Thermopylae and one at sea in Salamis. This puts the dinner of Esther and Aman, and the start of the festival of Purim, in year 475 BCE.
But 13 Adar is also the "Day of Nicanor", named after an enemy Seleucid general who oppressed the Judeans at the time of the revolt of the Maccabees. This Nicanor died in the battle of Adasa in 161 BCE against Judah Maccabee on 13 Adar. Here is what happened to him at the battle: Afterwards, they took the spoils, and the prey, and smote off Nicanor's head, and his right hand, which he stretched out so proudly; and brought them away and hanged them up towards Jerusalem. For this cause the people rejoiced greatly and they kept that day a day of great gladness. Moreover, they ordained to keep yearly this day, being the thirteenth of Adar. Thus, the land of Judah was in rest a little while. (source: Second Book of the Maccabees, 8:47-50)
Purim falls in the last month of the Hebrew calendar and is also the last festival of the Hebrew calendar. As mentioned above, this festival will also be the last one to remain after all other festivals will be ceased, in Messianic times. As such, Purim is often seen as the synthesis of all the festivals put together, and this adds to its importance. This is also because Purim has all the symbols of the other festivals. Indeed, it is about redemption from sure extermination (like Passover is with the Exodus and redemption from slavery). It is also about divine judgment with life or death (like Rosh Hashana, the Jewish New Year, where everyone is judged). It is also a day of Fast (like Kippur Day). It is also a day of Joy and sharing food with one another (like Sukkot). Strangely or not, the Book of Esther is the only book of the Bible where there is no mention of God at all, although divine designs can be perceived throughout. This adds to the importance of Purim festival because, in these times, the Jews brought salvation upon themselves, by their own merit and without the guidance or command of a prophet.
The word Purim is written פורים in Hebrew: its numerical value is 80+6+200+10+40= 336. And 336 = 4*12*7, where 4 is the number of seasons, 12 the number of months, and 7 the number of days of the week. So, we can see that Purim is also the synthesis of any calendar which is based on seasons/years, months, days/weeks. Furthermore 336 also equals to 2*24*7 which alludes to 7 days in a week, 24 hours in a day, and 2 being daylight and night each day. So, here also the Purim implies the human counting of time.
Many commentators brought to light a parallel between Purim and the Holocaust. For example, the Biblical text mentions the hanging of the 10 sons of Aman who conspired with him. And, after the trial of Nuremberg in 1946, 10 ministers and high dignitaries of the Nazi regime were condemned to death. And their execution was also performed by hanging.
Not the least, this parallel was drawn by the infamous Nazi propagandist Julius Streicher. Because, on 10 November 1938, the day after Kristallnacht, he declared that just as "the Jews butchered 75,000 Persians in one night, the same fate would have fallen over the German people had the Jews succeeded in inciting a war against Germany; the Jews would have instituted a new Purim festival in Germany." This same Streicher was condemned to death and executed in October 1946 after the Nuremberg Trial. On his way up to the scaffold, he shouted PurimFest 1946 (meaning: Purim Festival 1946). From Kristallnacht to his death, the festival of Purim must really have been an obsession.
Today, Purim is a very joyful festival. Many people dress up with fancy costumes. There are competitions about the best dress, and carnivals are organised in many cities. It is frequent to board on a bus or the tram in Israel where half the passengers are in costumes, and the driver as well ! It is truly a feast for the eyes.
Sadly, in year 2021, such displays are very restricted, or cancelled, due to the Covid crisis. But the Joy of Purim will not go away.