The Apple III has two different ways ways of accessing beyond the 6502's 64K addressing limit. The first is a fairly conventional bank-switching technique. The Apple III's memory is divided into three areas--a "lower system bank" from $0000 to $1FFF, an "upper system bank" from $A000 to $FFFF, and a bank-switched area in between (from $2000 to $9FFF). The two "system banks" are constant, and the in-between area can be bank-switched between as many as 15 32K banks. The four low-order bits at $FFEF control which bank is currently switched in--any number from $0 to $E can be specified ($F is reserved for some unspecified "special purpose"). Not all fifteen banks may be available--the highest available bank number in a 128K machine is 3, and the highest available bank in a 256K machine is 7. The second method is a bizarre thing called "extended addressing." This depends on a couple of other bizarre facts:
$00 through $FF. The 6502 is separated from memory by a "switch box" which (among other things) notices whenever an address of the form $00xx is referenced, and fetches (or writes to) memory location $nnxx instead, where nn is the current contents of memory location $FFD0. This means zero page can by moved anywhere in memory by storing a new address at $FFD0.$xxxx is accessed, the above-mentioned "switch box" also fetches the byte at "$xxxx XOR $0C00". (This extra byte is used by the video circuitry in 80-column mode.)
Whenever the 6502 uses a (ZP),Y addressing mode, and the zero page register ($FFD0) contains a number from $18 to $1F, the "switch box" uses the above-mentioned extra byte (called the "Xbyte") to perform "extended addressing." If the Xbyte (fetched from page ($FFD0) XOR $0C) is outside the range $80...$8F, then the indirect memory access occurs normally. Otherwise the switch box intervenes again and briefly remaps memory just long enough enough for the indirect memory access. If the low nibble of the Xbyte is n, then the indirect addressing sees a memory map composed of bank n (that's the same bank n from the normal bank-switching method mentioned above) mapped into $0000...$7FFF, and bank n+1 mapped into $8000...$FFFF.
....Well, almost. If the Xbyte is $8F, something special happens. The indirect addressing sees the normal Apple III memory map with bank 0 switched into $2000...$9FFF. As an added bonus, it also sees the RAM that's hiding under the VIA control registers at $FFD0...$FFEF. In fact, this is apparently the ONLY way to get at those 32 bytes of RAM. Since this memory is unavailable by other means, the operating system uses it to store the last valid value of the system clock.