In Judaism a girl reaches the status of becoming a Bat Mitzvah (literally, “responsible for observing commandments”) when she reaches the age of 12 years + one day, and a boy when he reaches the age of 13 years + one day. Traditionally, then, the service/ceremony celebrating this event would take place on the first day the Torah is read (Monday, Thursday, holiday and Shabbat mornings, as well as Shabbat afternoons) just following the child’s 12th or 13th birthday + one day.
Those who mark their birthdays according to the Hebrew calendar, would continue to do so for this event, while those who mark their birthdays according to the secular calendar would look at that date. Some families feel that though they almost always use the secular calendar (except for Jewish holidays, of course) for this religious event they want to have the ceremony on, or close, to the relevant Hebrew date. All of these options are fine.
In fact, unless one comes from a congregation which strictly observes the specific birthdate + one day tradition, many people today select a date close to the child’s birthday, but not necessarily immediately after it. Most people – especially those planning a trip to Israel – select a date which is convenient after taking into consideration school holiday schedules and the availability of friends and family members who will be joining them for this happy occasion.