The primary example is the Intel X86 architecture.
The Intel 8086 was, internally, a 16-bit processor: all of its registers were 16 bits wide. However, the address bus was 20 bits wide (1 MiB). This meant that you couldn't hold an entire address in a register, limiting you to the first 64 kiB.
Intel's solution was to create 16-bit "segment registers" whose contents would be shifted left four bits and added to the address. For example:
DS ("Data Segment") register: 1234 h
DX ("D eXtended") register: + 5678h
------
Actual address read: 179B8h
This created the concept of 64 kiB segment. Thus a "near" pointer would just be the contents of the DX register (5678h), and would be invalid unless the DS register was already set correctly, while a "far" pointer was 32 bits (12345678h, DS followed by DX) and would always work (but was slower since you had to load two registers and then restore the DS register when done).
(As supercat notes below, an offset to DX that overflowed would "roll over" before being added to DS to get the final address. This allowed 16-bit offsets to access any address in the 64 kiB segment, not just the part that was ± 32 kiB from where DX pointed, as is done in other architectures with 16-bit relative offset addressing in some instructions.)
However, note that you could have two "far" pointers that are different values but point to the same address. For example, the far pointer 100079B8h points to the same place as 12345678h. Thus, pointer comparison on far pointers was an invalid operation: the pointers could differ, but still point to the same place.
This was where I decided that Macs (with Motorola 68000 processors at the time) weren't so bad after all, so I missed out on huge pointers. IIRC, they were just far pointers that guaranteed that all the overlapping bits in the segment registers were 0's, as in the second example.
Motorola didn't have this problem with their 6800 series of processors, since they were limited to 64 kiB, When they created the 68000 architecture, they went straight to 32 bit registers, and thus never had need for near, far, or huge pointers. (Instead, their problem was that only the bottom 24 bits of the address actually mattered, so some programmers (notoriously Apple) would use the high 8 bits as "pointer flags", causing problems when address buses expanded to 32 bits (4 GiB).)
Linus Torvalds just held out until the 80386, which offered a "protected mode" where the addresses were 32 bits, and the segment registers were the high half of the address, and no addition was needed, and wrote Linux from the outset to use protected mode only, no weird segment stuff, and that's why you don't have near and far pointer support in Linux (and why no company designing a new architecture will ever go back to them if they want Linux support). And they ate Robin's minstrels, and there was much rejoicing. (Yay...)
