I've just done this in Drupal 7. Just install and enable the Date module, add a Date field to your Content Type, and then when editing it's settings, under "Date attributes to collect", just tick Year and Month. Then when a user is creating a new instance of that type, they will just get 2 dropdown boxes; one for Month and one for Year.
As for displaying them nicely, first go into Configutation > Date and Time settings, then create a new Format using PHP Date formatting e.g. "F Y" will give you something like "April 2012". Create a new Date Type which uses that format, and then go back to your Content Type display settings, and edit your new Date field to use the new Date Type.