Is a year really 365.24 days, or is it 365.2564 days like I remember?
Both are correct.

The Sidereal Year is the length of time it takes Earth to complete an orbit around the sun, relative to the fixed stars. It is 365.2564 days.

The Tropical Year is the length of time it takes for the Sun to complete a cycle around the Ecliptic and return to the position in the cycle of seasons; e.g. from Vernal Equinox to Vernal Equinox.  It's about 365.24217 days, about 20 minutes shorter than the Sidereal year because of the precession of the equinoxes. This is the year that the Gregorian Calendar is attempting to emulate.