You need to know about this, but it can be very complicated. If you stick to simple basics and don't try to get fancy, you will have no trouble with the colors you use on your website. All you need to know is that some colors look terrible and some look fine on just about everyone's computer. Just because you are an artist and work with colors everyday, you will still need to look carefully at how the colors you use for your business are represented on the web.
I don't advocate lots of graphics anyway, so you might want to simply get a professional web designer to make any custom graphics for you so you won't have to worry about this issue too much. Some people waste lots of time and money trying to match their web colors to some fancy custom brochure they have. It just doesn't work that way. Monitors cannot display accurately or even consistently colors as well as printing can.
If you really want to try it, do your web first and then drive your printer crazy trying to match ink to a website that looks different on every single monitor it is viewed on. Another important thing is the color scheme for your site. You don't want a website that is just screaming to people that you are trying to sell something. One of the main things you want to do is make people comfortable at your site. The colors that make people comfortable, are the colors they are used to seeing. Many people are new to this. They're scared to be on the computer in the first place. You might consider reducing your color scheme to a gray and blue, which is used in many Microsoft programs. People are accustomed to seeing these colors and it doesn't scream at you and don't forget about using only web safe colors as mentioned above.
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