The Hebrew Calendar is Lunar based.  The New Moon notes the first day of the month.  Occasionally, the "official" calendar deviates
from the actual New Moon.  April through October of this year is a good example.
HEBREW CALENDAR GREGORIAN CALENDAR
RED INDICATES DATES OF NOTE BLUE INDICATES NEW MOON
NISSAN - 5776 APR/MAY - 2016
S M T W T F S S M T W T F S
28 29 1 7 8 9
2 3 4 5 6 7 8 10 11 12 13 14 15 16
9 10 11 12 13 14 15 17 18 19 20 21 22 23
16 17 18 19 20 21 22 24 25 26 27 28 29 30
23 24 25 26 27 28 29 MAY 1 2 3 4 5 6 7
30 8
IYAR - 5776 MAY/JUNE - 2016
S M T W T F S S M T W T F S
1 2 3 4 5 6 9 10 11 12 13 14
7 8 9 10 11 12 13 15 16 17 18 19 20 21
14 15 16 17 18 19 20 22 23 24 25 26 27 28
21 22 23 24 25 26 27 29 30 31 JUNE 1 2 3 4
28 29 5 6
SIVAN - 5776 JUNE/JULY - 2016
S M T W T F S S M T W T F S
1 2 3 4 5 7 8 9 10 11
6 7 8 9 10 11 12 12 13 14 15 16 17 18
13 14 15 16 17 18 19 19 20 21 22 23 24 25
20 21 22 23 24 25 26 26 27 28 29 30 JULY 1 2
27 28 29 30 3 4 5 6
TAMMUZ - 5776 JULY/AUG - 2016
S M T W T F S S M T W T F S
1 2 3 7 8 9
4 5 6 7 8 9 10 10 11 12 13 14 15 16
11 12 13 14 15 16 17 17 18 19 20 21 22 23
18 19 20 21 22 23 24 24 25 26 27 28 29 30
25 26 27 28 29 31 AUG 1 2 3 4
AV - 5776 AUG/SEPT - 2016
S M T W T F S S M T W T F S
1 2 5 6
3 4 5 6 7 8 9 7 8 9 10 11 12 13
10 11 12 13 14 15 16 14 15 16 17 18 19 20
17 18 19 20 21 22 23 21 22 23 24 25 26 27
24 25 26 27 28 29 30 28 29 30 31 SEPT 1 2 3
ELUL - 5776 SEPT/OCT - 2016
S M T W T F S S M T W T F S
1 2 3 4 5 6 7 4 5 6 7 8 9 10
8 9 10 11 12 13 14 11 12 13 14 15 16 17
15 16 17 18 19 20 21 18 19 20 21 22 23 24
22 23 24 25 26 27 28 25 26 27 28 29 30 OCT 1
29 2
TISHRI - 5777 OCT/NOV - 2016
S M T W T F S S M T W T F S
1 2 3 4 5 6 3 4 5 6 7 8
7 8 9 10 11 12 13 9 10 11 12 13 14 15
14 15 16 17 18 19 20 16 17 18 19 20 21 22
21 22 23 24 25 26 27 23 24 25 26 27 28 29
28 29 30 30 31 NOV 1
HESHVAN - 5777 NOV - 2016
S M T W T F S S M T W T F S
1 2 3 4 2 3 4 5
5 6 7 8 9 10 11 6 7 8 9 10 11 12
12 13 14 15 16 17 18 13 14 15 16 17 18 19
19 20 21 22 23 24 25 20 21 22 23 24 25 26
26 27 28 29 27 28 29 30
KISLEV - 5777 DEC - 2016
S M T W T F S S M T W T F S
1 2 3 DEC 1 2 3
4 5 6 7 8 9 10 4 5 6 7 8 9 10
11 12 13 14 15 16 17 11 12 13 14 15 16 17
18 19 20 21 22 23 24 18 19 20 21 22 23 24
25 26 27 28 29 25 26 27 28 29
TEVETH - 5777 DEC/JAN - 2017
S M T W T F S S M T W T F S
1 2 30 31
3 4 5 6 7 8 9 JAN 1 2 3 4 5 6 7
10 11 12 13 14 15 16 8 9 10 11 12 13 14
17 18 19 20 21 22 23 15 16 17 18 19 20 21
24 25 26 27 28 29 22 23 24 25 26 27
SHEBAT - 5777 JAN/FEB - 2017
S M T W T F S S M T W T F S
1 28
2 3 4 5 6 7 8 29 30 31 FEB 1 2 3 4
9 10 11 12 13 14 15 5 6 7 8 9 10 11
16 17 18 19 20 21 22 12 13 14 15 16 17 18
23 24 25 26 27 28 29 19 20 21 22 23 24 25
30 26
ADAR - 5777 FEB/MAR - 2017
S M T W T F S S M T W T F S
1 2 3 4 5 6 27 28 MAR 1 2 3 4
7 8 9 10 11 12 13 5 6 7 8 9 10 11
14 15 16 17 18 19 20 12 13 14 15 16 17 18
21 22 23 24 25 26 27 19 20 21 22 23 24 25
28 29 1 26 27 28
Nisan 1 Correct beginning of the year - Exodus 12:1-6
Nisan 10 Passover Lamb chosen (Exodus 12:3)
Nisan 14 Passover begins at Sundown (Leviticus 23:5)
Nisan 15 The Feast of Unleavened Bread (Leviticus 23:6)
Nisan 16 The first Sunday after Passover is the Day of Firstfruits
and begins the fifty day count for Pentecost  (Leviticus 23:11)
Sivan 6 Shavuot - Pentecost (Leviticus 23:16)
Tammuz 17 Traditional anniversary of destruction of the walls of Jerusalem (twice)
Av 9 Traditional anniversary of destruction of the Temple (twice)
Tishri 1 Rosh HaShanah - Feast of Trumpets (Leviticus 23:23)
Tishri 10 Yom Kippur - Day of Atonement (Leviticus 23:26)
Tishri 15 Sukkot - Feast of Booths, Feast of Harvest (Leviticus 23:33)
Kislev 25 Chanukah - Festival of Lights (see History of Maccabean Revolt)
Adar 14 & 15 Purim - see Book of Esther