In the cells you want the month in, start with January and hold the left mouse button down while dragging the cursor in whichever direction you want the months to go. It will loop back to January once you hit December.
Type "January" in any cell. Drag the small box located in the bottom right corner of the cell cursor down. Remaining cells will be filled up with the remaining months.
Select the cell you wish to contain the current month. In that cell enter the following:
=today()
Note that the cell now contains the full date. To have it contain only the month right click the cell and select "Format cell". Select the data category and choose the entry that has the month's name.
In order to show just the month of the current date, you can use the following Excel formula: =month(now()). Using the now() function inside of the month() function returns the number value for the month of today's date.