One of the challenges of being a web designer is that no matter how much care you take with your design, your page is going to be viewed on other people's systems. When I started working in web design in 1995 that meant designing for people with black and white or gray scale monitors, people with monitors with only a few colors (256 or fewer in many cases), and even people on monitors that couldn't display anything but text. These days most desktop monitors have millions of colors, so worrying about contrast on a gray scale monitor is less important. But that doesn't mean you're off the hook. If your monitor is not correctly calibrated (and most are not, even right out of the box) the colors you view could be vastly different from what your customers see.
This article from the About.com Desktop Publishing Guide will help you calibrate your monitor: Calibrate Your Monitor. But remember that once you get your monitor showing colors correctly, that doesn't mean that your customers will have correctly calibrated monitors. This article, Web Color Synchronization gives suggestions for how to deal with getting good colors even if your customers have poorly calibrated monitors.
Learn More: Color Calibrate All Your Devices
Image courtesy foobean01 from StockXchng #537438.