I think that all the Intel chips I've ever bought were IA-32 architecture.
Are Intel Itaniums sold in desktops or laptops?
Itanium was a server play for Intel. It was a way to shed the history of IA32 and try a brand new architecture, a design called EPIC, for Explicitly Parallel Instruction Computing. Some of the early design inspiration was based on HPs PA-RISC architecture and they worked with HP in overall design. Intel wanted to copy what they did with IA32, have a common chip for all big servers and leverage massive economies of scale. Since you're asking what Itanium is, they obviously didn't do very well on the scale part :). Its nickname was Itanic, obviously not a name that indicates massive economic success.
As far as the chip market, it did accomplish one thing. It scared some other RISC vendors out of the market. Part of the reason for SGI dropping MIPS as a workstation chip and DEC dropping Alpha was the threat of Itanium. They figured if Intel can dominate with the badly designed IA32 architecture, what if they had a clean slate and money to back it up? They dropped out, figuring they'd port to Itanium and still sell their OS. SGI soldiered on a bit selling Itanium workstations, but their ability to be different was smashed, and they died soon after (which makes you congratulate Apple a bit being able to sell near commodity Intel laptops). HP just wanted to cut their chip costs down (they were making both Alpha and PA-RISC) and instead concentrate on moving printer ink.
As far as actually selling Itaniums in the market, it kind of landed with a thud. The new EPIC architecure made it VERY compiler dependent, and there were no good compilers in the beginning (and maybe not even now). It had the classic chicken and egg problem - no apps because no systems sold, no systems sold because of no apps. And its IA32 support sucked in the beginning. The first versions of the chip were particularly bad, though got a bit better on later generations.
Eventually, AMD released 64 bit extensions to IA32, x86_64, AMD64, whatever you want to call it. This gave decent speed at not a huge cost jump. The internal architecture was easy to write compilers for, and had very good IA32 performance. It cleaned up. Intel was forced to backtrack, and released the extensions as EM64T. It had the rights to from earlier licensing agreements with AMD regarding 486 production. Itanium would be forever relegated to a niche server product.
As far as "What has replaced them", nothing really. Itanium didn't really sell well, but it's still being produced. If you actually have Itanium, you can replace with a newer one if you like. If not, there are no emulators that I know of, you'd need to port your code to a new architecture. If it's fast enough for you, Intel Xeon (server versions of normal Intel chips), if not, probably IBM POWER. But you'd need to buy new machines.
And you probably don't buy IA32 chips anymore, you most likely buy EM64T chips, which have great IA32 compatibility.
TL;DR: It was a big-iron server chip, that never sold well, and cheaper 64 bit Intel chips (x86_64, EM64T, whatever you want to call them) took much of its reason to be.
They're pretty much exclusively found in server environments, and not particularly often these days either.