For photographic editing, wide gamut is a good thing - it means your screen can display more of the original captured colours accurately
A wide gamut monitor is a monitor that can display more colours than a standard monitor - and by more colours, we are mostly referring to stronger, more saturated colours. A standard monitor (there is not really a 'standard' but think of your older normal desktop monitors in this instance) - can display about 75% of the AdobeRGB colour space, or roughly the same gamut as sRGB. sRGB was in fact originally designed as a colour space describing the gamut normal monitors would typically display.
Most monitors currently being sold are wide gamut monitors - it's not
just NEC and Eizo, but basically all the manufacturers are moving to
wide gamut - and this will only increase, as there is a general push
from the panel foundries toward wide gamut displays, both for computer
usage and for televisions and public displays.
For photographic editing, wide gamut is a good thing - it means your screen can display more of the original captured colours accurately. Particularly with saturated images, like for example late afternoon landscapes, it can often make a significant difference in the display quality. Further, it means your monitor can accurately display more of the colours your printer can print - meaning overall improved screen to print matches.
Wide gamut can cause some issues. Most problems arise because your computer doesn't really understand that your new monitor is wide gamut - most of the software has been designed with traditional monitors in mind, as wide gamut screens are a relatively new development.
Your operating system (be it Windows or OSX or anything else) - will generally behave as if a standard gamut screen is attached, and send signals appropriate for standard gamut screens. So the basic problem is one of signal - the signal going to your wide gamut monitor is designed to trigger roughly the appropriate colours on a standard gamut monitor. This same signal sent to a wide gamut monitor will produce much stronger colour.
A simple example of this is the icons on your desktop. These are very basic graphic files, often designed years ago, and have only basic colour numbers in them. The operating system does not colour manage these numbers and sends them directly to the screen as they are.
Say you have Firefox installed - the little red fox in the icon contains some very strong reds - these are actually defined as Red 255 in the file. When sent to any monitor, Red 255 means 'display your strongest red'. On a standard gamut monitor, the strongest red is quite strong. On a modern wide gamut monitor, this red will be much stronger again, and may well appear quite over-saturated.
This problem/principle will apply to any numbers being sent to your screen in a raw, non colour managed form. By and large, the problems are minor and not of great practical importance - the day to day experience is often just that a few things look over-saturated.
The problem can be solved by modifying the outgoing signal from your video card to be correct for wide gamut images - that is, applying monitor colour management using the monitor's profile. So the trick is to get as much of the signal to be colour managed as possible.
The first thing you really must do is calibrate your wide gamut monitor using a hardware calibration device - this process produces an ICC profile that colour managed applications can use to correct the outgoing signal when displaying images - without doing this, you will never get your wide gamut monitor displaying things correctly. The only exception to this is if you can put your wide gamut monitor into a smaller gamut mode (usually labelled 'sRGB emulation' mode). Most of the newer monitors simulate sRGB well, and most can set sRGB as the gamut for a calibration target, meaning you can have a calibrated sRGB scenario.
Assuming you have a wide gamut monitor running in its native gamut mode, and you've properly calibrated it using a hardware device, let's now look at some specific scenarios and see if there are solutions to the issues.
The operating system in general, and the basic applications that come with it, will generally send only a raw signal to your screen and will thus display over the top colour. Windows and Mac differ here slightly - more of the built in Mac applications are colour managed than on the PC, but the Mac desktop itself isn't, and is very luridly coloured. In general the Mac has a way of being both worse and a little bit better - the desktop looks generally heavily over-saturated, but the images inside colour managed applications, including the built in image viewer called Preview, will look correct (assuming you've calibrated your monitor of course!). The PC interface is a bit more sedate so doesn't look as bad overall, but the PC does not come with an image viewer that is colour managed by default, so you need extra software to solve the problem on the PC (see below).
In the case of your operating system, general apps, and desktop icons, there's currently no proper solution - it's likely that future updates to the OS will either fix this problem by colour managing the entire desktop, or more likely the icons will be re-designed with a less intense palette as wide gamut monitors become increasingly ubiquitous.
Most modern imaging programs support colour management. On the Mac, this includes the built-in Preview app, and pretty much all the serious image editing software (although iPhoto's implementation of colour management in notoriously unreliable at best). Certainly Photoshop/Elements/Lightroom/Aperture and the like are colour managed. All of these colour managed apps will display images correctly (once you've calibrated your monitor).
On the PC, the built in Windows Picture and Fax viewer is NOT currently colour managed and will display images incorrectly on wide gamut screens. You must use after-market software which supports colour management to solve the issue - some are free such as Zoner Photo Studio Free, and others are commercial like the classic ACDSee. There are literally hundreds of options in this area. All of these apps can use a monitor profile to send images correctly to calibrated wide gamut screens. Once installed, these can completely replace the in-built Windows viewer and are much better software in general to boot, so it's well worth the effort. All of the serious apps such as Photoshop/Elements/Lightroom etc are of course fully colour managed as well.
By default, most web browsers are not colour managed. We have an article explaining the state of colour management and web browsers. The short version is, use Firefox and the colour management add on to turn on fully colour managed browsing. Once you are using a colour managed browser, you can browse the web as normal and in almost all cases images will look as they should.