I know at least 2 ways. Open a terminal and type:
uname -a
Result for 32-bit Ubuntu:
Linux discworld 2.6.38-8-generic #42-Ubuntu SMP Mon Apr 11 03:31:50 UTC 2011 i686 i686 i386 GNU/Linux
whereas the 64-bit Ubuntu will show:
Linux discworld 2.6.38-8-generic #42-Ubuntu SMP Mon Apr 11 03:31:50 UTC 2011 x86_64 x86_64 x86_64 GNU/Linux
or
file /sbin/init
Result for 32-bit Ubuntu:
/sbin/init: ELF 32-bit LSB shared object, Intel 80386, version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.6.15, stripped
whereas for the 64-bit version it would look like:
/sbin/init: ELF 64-bit LSB shared object, x86-64, version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.6.15, stripped