Where are Intel Itanium chips used?
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.
EDIT So now even HP, one of the architects of the Itanium is now even slowly migrating away from Itanium, moving to Xeon x86_64 chips.
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.
LAST EDIT Annnnnddddd they're dead
They're pretty much exclusively found in server environments, and not particularly often these days either.
Itanium was a processor architecture intended for use in servers and high-performance computing. It aimed to deliver high performance and high efficiency by implementing a special type of very long instruction word (VLIW) architecture called Explicitly Parallel Instruction Computing (EPIC).
In order to maximize performance, conventional processors contain lots of circuitry for scheduling instructions so that they can run as many instructions as possible in parallel (see this answer for a detailed technical discussion on the techniques used to do this). VLIW, and in particular EPIC, attempted to eliminate this complex, power-consuming circuitry by shifting this work to the compiler. The compiler would determine what instructions can be run in parallel by the various execution units in the processor and code this information directly into the executable binary (hence "explicitly parallel"). In essence, it was hoped that improving compiler technology could be used in place of hardware instruction scheduling circuitry, allowing for more efficient and more scalable processors.
Unfortunately, the compiler technology needed to utilize this new architecture to its fullest never materialized. Many scheduling optimizations made by the processor at runtime proved to be (and still are) very difficult to do at compile time. The result was poor performance that could not keep up with more conventional x86, Power, and SPARC servers.
Furthermore, the Itanium architecture is not compatible with x86. While Intel had implemented an x86 compatibility layer, performance was very poor compared to an actual x86 processor. On the other hand, AMD developed an extended version of x86 called AMD64 (now x86-64) that is backwards-compatible with x86 systems and gained much wider acceptance. Today, x86-64 is the architecture used in the vast majority of PC and server processors, while Itanium proved to be a commercial failure and has since fallen into obscurity.