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.

It means that you're running a 32-bit kernel, which means that you can only run 32-bit apps without the use of an emulator or virtualization.

If you want to see if your processor is 64-bit then look for lm in the flags listed in /proc/cpuinfo.

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.