Googles stock contact app (the contact application found on Nexus one or Nexus S) does show a contacts birthday but does not allow to edit it. If you want to edit the contacts birthday you'll have to do it in gmail. This application provides a mean to add, edit and remove the birthdays of contacts originating from your google account.
In addition to this it can send a notification of someones birthday to the status bar. The reminder service can be activated and deactivated and the time of the notification can be set.