OK, this may take a while to explain. This passage in Bad Astronomy is on page 55:
The last two sentences are in error, the key phrase is "our current calendar".Right now, the Earth's north axis points toward the Sun in June. But due to precession, 13,000 years from now--half a precession cycle--the Earth's north pole will be pointed away from the Sun in June and toward it in December. Seasons will be reversed relative to our current calendar.
The Earth actually takes slightly more than 365.25 days to go around the Sun. Early calendars accounted for the extra quarter day or so, but because of precession, the date of the vernal equinox gradually changed. The first day of Spring was off by more than ten days, after a few hundred years. That effect is what is described in the Bad Astronomy book.
However, Pope Gregory decreed that the vernal equinox was to always occur late in March, and so they skipped the extra days to bring it back into place, and defined the length of the year to just a little bit less than 365.25 days. That is called the tropical year (or the astronomical year), and it is what our current calendar (the Gregorian calendar) is based upon. It is the time from one vernal equinox to the next.
It is defined that way so that the vernal equinox (and therefore the summer solstice) will always occur at the same time of the year, and so the Earth's north pole will always tilt towards the Sun in June.