What do x86_64, i386, ia64 and other such jargons stand for?

I frequently encounter these terms and am confused about them. Are they specific to the Processor, or the Operating System, or both?

I have Ubuntu 12.04 running on Intel i7 machine. So which one of them would apply for my case?

They are processor instruction set names:

• i386 is the name of the 32-bit instruction set first implemented by Intel in the 386 processor. It became dominant thanks to dirt-cheap PC hardware.
• x86-64 is the name of the AMD extension added to i386 to make it capable of executing 64-bit code. This is the one you have. It is highly compatible with i386 and will execute a 32-bit program as fast as an i386 processor.
• ia64 is the name of the instruction set used in Itanium processors. The other 64-bit architecture that nobody uses anymore.