Assuming you have a column named "Month", you will need to add quotes around each month name. (They are just text.)
=IF([Month]="JAN","1",
IF([Month]="FEB","2",
IF([Month]="MAR","3",
IF([Month]="APR","4",
IF([Month]="MAY","5",
IF([Month]="JUN","6",
IF([Month]="JUL","7",
IF([Month]="AUG","8",
IF([Month]="SEP","9",
IF([Month]="OCT","10",
IF([Month]="NOV","11",
IF([Month]="DEC","12",
""
))))))))))))
If you want the result as a number, then remove the quotes from the numbers:
=IF([Month]="JAN",1,
IF([Month]="FEB",2,
IF([Month]="MAR",3,
IF([Month]="APR",4,
IF([Month]="MAY",5,
IF([Month]="JUN",6,
IF([Month]="JUL",7,
IF([Month]="AUG",8,
IF([Month]="SEP",9,
IF([Month]="OCT",10,
IF([Month]="NOV",11,
IF([Month]="DEC",12,
0
))))))))))))
Here's a much short formula that does the same thing:
=(FIND(UPPER(D2)," ,JAN,FEB,MAR,APR,MAY,JUN,JUL,AUG,SEP,OCT,NOV,DEC")-1)/4
(There are three spaces between the quote and the first comma.)
And here's one that will work with any three to nine letter month name:
=(FIND(UPPER(Month)," ,JANUARY ,FEBRUARY ,MARCH ,APRIL ,MAY ,JUNE ,JULY ,AUGUST ,SEPTEMBER,OCTOBER ,NOVEMBER ,DECEMBER ")-1)/10
(All of the spaces are important!)