The Endian-ism of a processor refers to the order in which multi-byte values are stored in memory.
Little-endian systems store the least-significant byte of a multi-byte value at the lowest address.
For example, on 6502 systems (with a 8 bit/1 byte word size), the 16-bit value $FFEE stored at address $1000 would be stored like this:
$1000 $EE $1001 $FF
Likewise, on an x86_64 system, the 64-bit value 0xFFEEDDCCBBAA9988 would be stored at memory location 0x1000 like this:
0x1000 0x88 0x1001 0x99 0x1002 0xAA 0x1003 0xBB 0x1004 0xCC 0x1005 0xDD 0x1006 0xEE 0x1007 0xFF
Big-endian systems store multi-byte values with the most significant byte at the lowest address.
For example, on a Motorola 68030 CPU, the value 0xFFEEDDCC would be written at memory location 0x1000 as:
0x1000 0xFF 0x1001 0xEE 0x1002 0xDD 0x1003 0xCC
Other Memory Sequences
There are a few, rare architectures which store values in a sequence other than strict little-endian or big-endian format. For example, the 32-bit value FFEEDDCCh would be stored by a PDP-11 at address 1000h in this manner:
1000h EEh 1001h FFh 1002h CCh 1003h DDh
That is, each 16-bit word is stored in little-endian format, but the two 16-bit words are in big-endian sequence.
Significance of Byte Order
Byte order can introduce ambiguity and complexity when transferring binary data between dissimilar systems; in network protocols and file specifications, it is critical to specify the order of multi-byte values. Some platform-independent remote procedure call (RPC) systems provide functions to ensure that data is always represented "on the wire" in a particular byte order.
Byte order can be significant for performance in some contexts. The 6502 processor, for example, loads the byte following an opcode in memory while the opcode is being decoded. Due to little-endian encoding, this byte can be safely assumed to represent bit positions 0-7 regardless of whether the opcode's argument is 1 byte or 2 bytes; if big-endian encoding was used, the first byte read after the opcode might have to be shifted to bit positions 8-15 in the case of a 2-byte argument.
Binary fields in internet protocol (IP) packets are in big-endian order. For this reason, the processors on many routers and other high-speed network devices are big-endian (e.g., MIPS) or run in big-endian order (ARM).
Origin of the Terms
The terms Big-Endian and Little-Endian originally appeared in the classic satirical tale Gulliver's Travels by Jonathan Swift, where they referred to which end of a boiled egg was to be opened first -- a distinction which led to ongoing conflict between small-minded people.