Welcome to the blurry web
Currently, the vast majority of web developers are still building new websites that are not ready for retina displays. If you look closely at most sites using a device with high density retina display, you’ll notice that the graphics and imagery will look blurry. Something that such device owners frequently complain about.
A retina display has such a high density of pixels that the human eye cannot see the pixel definition within the graphics at all, which allows for the clearest and sharpest graphics possible.
It's important to note
The vast majority of your audience will not be experiencing your website on a device with retina display, as the capability to display such high resolution graphics only applies to the latest Apple products (iPhone, iPad, Macbook Pro & Macbook Air).
If only Apple products, why bother?
Over 50 million users have already purchased these Apple devices, that should be a good enough reason. However, I'm sure you would agree that over the last few years, Apple really has been the front-runner in device evolution and where they go others tend to follow. So it shouldn't be long before retina displays are considered "standard" for any home/office computers, laptops and tablet devices.
Therefore, if you are considering updating your website. It would be highly beneficial to make sure that "retina ready" is in bold on your list of requirements.