Get Month
Month of the Year
Use =MONTH as in =MONTH(A1) to get the month of a date entered in A1.
Month Name
Formatting as mmm or mmmm will return Jan or January so use either of these formula instead. (But it is okay for use with =TODAY() or =NOW() )
=CHOOSE(MONTH(A1),"Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec")
or
=CHOOSE(MONTH(A1),"January","February","March","April","May","June","July","August","September","October","November","December")
For the current month, =TODAY() or =NOW() formatted as mmm or mmmm is okay. (You can use the TEXT function as in =TEXT(TODAY(),"mmmm") also)