How can you tell whether code was compiled for IA-32 or x86-64?

Question

Accepted Answer

Read the register names and argument passing: %r8–%r15 or arguments arriving in %rdi/%rsi means x86-64; everything via the stack with %eXX registers means IA-32.
A few quick visual tells separate the two:

Feature
IA-32 (32-bit)
x86-64 (64-bit)

Registers
%eax, %ebx, %ecx…
%rax…, plus %r8–%r15

Arguments
all on the stack
first 6 in registers

Frame pointer
almost always %ebp
often omitted

Pointer size
4 bytes
8 bytes

Stack ops
pushl, popl
pushq, popq

IA-32 signature — arguments fetched as offsets from %ebp:
movl  8(%ebp), %eax
movl  12(%ebp), %edx
pushl %ebx

x86-64 signature — arguments already in registers, plus the new R-registers:
movq  %rdi, %rax
movq  %rsi, %rdx
movq  %r12, -8(%rsp)

Quickest check: spotting any %r8–%r15, or argument use of %rdi/%rsi, is a dead giveaway that you're looking at 64-bit code.
Go deeper:

x86-64 (Wikipedia) — contrasts the 8 IA-32 vs 16 x86-64 registers.

Feature	IA-32 (32-bit)	x86-64 (64-bit)
Registers	%eax, %ebx, %ecx…	%rax…, plus %r8–%r15
Arguments	all on the stack	first 6 in registers
Frame pointer	almost always %ebp	often omitted
Pointer size	4 bytes	8 bytes
Stack ops	pushl, popl	pushq, popq