The size_t type is defined as the unsigned integral type of the sizeof operator. In the real world, you will often see int defined as 32 bits (for backward compatibility) but size_t defined as 64 bits (so you can declare arrays and structures more than 4 GiB in size) on 64-bit platforms. If a long int is also 64-bits, this is called the LP64 convention; if long int is 32 bits but long long int and pointers are 64 bits, that’s LLP64. You also might get the reverse, a program that uses 64-bit instructions for speed, but 32-bit pointers to save memory. Also, int is signed and size_t is unsigned.
There were historically a number of other platforms where addresses were wider or shorter than the native size of int. In fact, in the ’70s and early ’80s, this was more common than not: all the popular 8-bit microcomputers had 8-bit registers and 16-bit addresses, and the transition between 16 and 32 bits also produced many machines that had addresses wider than their registers. I occasionally still see questions here about Borland Turbo C for MS-DOS, whose Huge memory mode had 20-bit addresses stored in 32 bits on a 16-bit CPU (but which could support the 32-bit instruction set of the 80386); the Motorola 68000 had a 16-bit ALU with 32-bit registers and addresses; there were IBM mainframes with 15-bit, 24-bit or 31-bit addresses. You also still see different ALU and address-bus sizes in embedded systems.
Any time int is smaller than size_t, and you try to store the size or offset of a very large file or object in an unsigned int, there is the possibility that it could overflow and cause a bug. With an int, there is also the possibility of getting a negative number. If an int or unsigned int is wider, the program will run correctly but waste memory.
You should generally use the correct type for the purpose if you want portability. A lot of people will recommend that you use signed math instead of unsigned (to avoid nasty, subtle bugs like 1U < -3). For that purpose, the standard library defines ptrdiff_t in <stddef.h> as the signed type of the result of subtracting a pointer from another.
That said, a workaround might be to bounds-check all addresses and offsets against INT_MAX and either 0 or INT_MIN as appropriate, and turn on the compiler warnings about comparing signed and unsigned quantities in case you miss any. You should always, always, always be checking your array accesses for overflow in C anyway.