Google Contacts allow setting birthdays, which are then picked up by Google Calendar automatically and displayed as events on the relevant day. This is very convenient for keeping track of your relatives', friends' and other people's birthdays. Having a reminder saves you an embarrassment of forgetting your girlfriend's/boyfriend's/auntie's day of birth and prevents unnecessary bad feelings.
For Android users it makes obvious sense to be able to see these events on their phones directly. In theory it should just work - in Calendar app choose "Calendars" menu, then select "Contacts' birthdays and events" from the list and you are done.
However, there seems to be a problem with Google Calendar synchronization. When you add a new contact's birthday, it may not get synchronized with your phone. A solution to this issue is not complicated: go into Calendar application again, unselect "Contacts' birthdays and events", accept the change, then go back and re-enable the calendar. This time it should synchronize properly.