Is my CPU 32 bits or 64 bits?

The operating system on my machine is Windows 7 32bit.

Although Windows lists as 32 bits, but does also mean my CPU is 32 bits? If not how could I find out what my CPU is?

The T6600 is a 64 bit CPU which means you can use it to run either the 32-bit or the 64-bit version of Windows. Its specifications say:

Instruction Set: 64-bit

Although your Windows is a 32-bit installation your CPU is capable of running a 64-bit operating system. This support is indicated by the existance of the Yes value for the Intel® 64 product feature on the Intel® Core™2 Duo Processor T6600 on the specification page within the Advanced Technologies section

The important "Instructions" flag that CPU-Z shows here is EM64T, which is the extension of Intel CPU's to execute 64 Bit commands (along with 32 Bit, which are still possible). On an AMD CPU the according flag would be x86-64.

What the Windows "system type" displays is the actual word width that is currently used. In your case 32 Bit.

Conclusion: your CPU is able to run 64 Bit Windows and 32 Bit Windows (what you have). If you want to run 64 Bit programs, you need 64 Bit Windows (which still can run 32 Bit applications). If you have a Windows 64Bit setup DVD/image or know someone that has one, you can install Windows 64 Bit with the key from your 32 Bit Windows (they are interchangable between 32 and 64 bit, as long as the key itself is valid and used within its license restictions). Backup you data, however, since upgrading 32 to 64 Bit is not possible (at least not straightforward), only a fresh install is possible when switching the architecture.