Is My Processor 64 or 32 Bit?

You can use lscpu.

someuser@somelaptop:~$ lscpu
Architecture:          i686           # <-- your kernel is 32 bit
CPU op-mode(s):        32-bit, 64-bit # <-- your cpu can handle 32 or 64 bit instructions
CPU(s):                4
Thread(s) per core:    2
Core(s) per socket:    2
CPU socket(s):         1
Vendor ID:             GenuineIntel
CPU family:            6
Model:                 37
Stepping:              5
CPU MHz:               1199.000
Virtualisation:        VT-x
L1d cache:             32K
L1i cache:             32K
L2 cache:              256K
L3 cache:              3072K

Further explanation of the Architecture field:

X86, i686, or i386      means you are running a 32 bit kernel.
X86_64 , amd64 , or X64 means you are running a 64 bit kernel.

You can also check the architecture of the binary you're trying to run by using file: file filetocheck. It will mention either 32-bit or 64-bit.

uname -p give the architecture of the processor. If it gives x86_64, it means cpu is 64 bit.

Basic idea:

x86_64 is 64 bit capable cpu and i386 is 32 bit.

With lscpu

Long answer: lscpu

Architecture:          x86_64
CPU op-mode(s):        32-bit, 64-bit
Byte Order:            Little Endian
CPU(s):                2
On-line CPU(s) list:   0,1
Thread(s) per core:    1
Core(s) per socket:    2
Socket(s):             1
NUMA node(s):          1
Vendor ID:             AuthenticAMD
CPU family:            17
Model:                 3
Stepping:              1
CPU MHz:               550.000
BogoMIPS:              4397.92
Virtualization:        AMD-V
L1d cache:             64K
L1i cache:             64K
L2 cache:              512K
NUMA node0 CPU(s):     0,1

Short answer lscpu | awk '/CPU op-mode/ {print}

Output : CPU op-mode(s): 32-bit, 64-bit

With lshw

Long answer: run sudo lshw

Slightly shorter answer: sudo lshw -c cpu

Output:

 *-cpu                   
       description: CPU
       product: AMD Turion(tm) X2 Dual-Core Mobile RM-75
       vendor: Advanced Micro Devices [AMD]
       physical id: 4
       bus info: cpu@0
       version: Turion X2 Mobile RM-75
       slot: Socket M2/S1G1
       size: 550MHz
       capacity: 4GHz
       width: 64 bits
       clock: 200MHz
       capabilities: fpu fpu_exception wp vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt rdtscp x86-64 3dnowext 3dnow constant_tsc rep_good nopl nonstop_tsc extd_apicid pni cx16 lahf_lm cmp_legacy svm extapic cr8_legacy 3dnowprefetch osvw skinit hw_pstate lbrv svm_lock nrip_save vmmcall cpufr

Even shorter answer: sudo lshw -c cpu | grep width

Output: width: 64 bits