How to use this table:

The year:

The Gregorian lunar almanac is a 19-year cycle which assigns an age of the moon to every day in the 19-year period.

Year one of the Gregorian cycle is any year in which the year's number is divisible by 19 without remainder.  The year 1786, for example, was year one of its cycle:  The table shows that Lunar Year 1786 began on January 1st, 1786.  (More precisely, at sunset on December 31st, 1785).   Successive years of the cycle follow in order.  The year 1787 was year two:  Lunar Year 1787 began on December 21st, 1786  (Or more precisely, at sunset on December 20th, 1786).   The year 1788 was year 3:  Lunar Year 1788 began on December 10th, 1787.
 

The month:

Each year contains 12 or 13 lunar months.  Each lunar month has 29 or 30 days.

The table shows the first day of each lunar month in the Gregorian Easter cycle.  The moon is considered to be one day old on the first day of the lunar month.  This day is also called "new moon" because it corresponds, on the average, to the day on which the new waxing  crescent moon theoretically first becomes visible.

If February has 29 days, an extra day is added to month 2, making it a month of 30 days, rather than 29 days.  Month 3 then begins a day later than the date listed in the table if the date listed is a date prior to February 29th.  (This is not the only  possible way of accounting for a 29-day February, but it is the easiest.)

The last lunar month of the 19th year of the cycle has 29 days.
 

The day:

Because this is a Babylonian-style lunar calendar, the day begins on sunset of the day prior to the day listed in the table.  The table shows that the first day of the first moon of year 1 of the cycle corresponds to January 2nd.  This means that the first day  of the lunar month begins at sunset on January 1st and ends at sunset on January 2nd, when the second day of the lunar month begins.
 

The Easter festival falls in the first lunar month of the year to begin on or after March 8th.  In the years 1700-1899, this is always the fourth moon.  Easter is always the third Sunday in its lunar month.

In the Earth's Northern hemisphere, the Spring season of year 4 of the Gregorian cycle coincides with the Spring season of year 1 of the Jewish 19-year cycle, with other years of each cycle following in succession.  The Feast of Unleavened Bread for the year 5565 (year 17 of the Jewish cycle) came in the Northern hemisphere's springtime in the Gregorian year 1805 (year 1 of the Gregorian cycle.)

Due to differences in the time of year at which the Christian and Jewish cycles add the 13th moon of 13-moon years, the Jewish Feast of Unleavened Bread, in years 3, 11, and 14  of the Gregorian cycles spanning the years 1700-1899 (corresponding respectively to years 19, 8, and 11 of the Jewish cycle), fell in the lunar month next after the lunar month in which the Christian Easter festival fell.  In all other years of the cycle, Easter and the Feast of Unleavened Bread fell in the same lunar month.  But though the lunar month in which the Jewish Feast of Unleavened Bread fell can be identified using the table above, the precise dates of the Feast of Unleavened Bread cannot be determined accurately from the table.  This is because the Gregorian and Jewish calendars use different rules for computing the moon's age.

References:

Explanatory Supplement to the Ephemeris and the American Ephemeris and Nautical Almanac, H. M. Stationery Office, London, 1966, fourth revised printing, 1977.

Alexander Philip, The Calendar: Its History, Structure and Improvement, University Press, Cambridge, 1921.

L. A. Resnikoff, "Jewish Calendar Calculations I", Scripta Mathematica 9, 191(1943); "Jewish Calendar Calculations II", Scripta Mathematica 9, 274(1943).