In layman's terms, the resolution of your computer monitor or screen determines how much website content can display on your screen. In technical terms, screen resolution represents the physical number of columns and rows of pixels displaying on the screen.
Designing a new law firm website or blog for the appropriate screen resolution of your typical user is important since if you design for a higher resolution (typically 1024 x 768), then users who have lower resolution screens (typically 800 x 600) are going to have to scroll horizontally to view content on the far righthand side of your website. That is annoying, and bad news for your firm if you have a significant percentage of users with 800 x 600 screens (note: the area of the browser window available for displaying content is almost always smaller than the full display resolution because some space is taken up by the browser frame, i.e., title, function bars, scroll bars, etc. - so generally, optimizing for, say, an 800x600 screen means a design that is 750-760 pixels wide).
There was a time when a significant percentage of Internet users (e.g., 35-40%) were using 800x600 screens (see screen resolution percentage breakdowns in 2002). Since 2002, the percentage of 800x600 screen has steadily decreased, and nowadays the consensus is that close to 90% of Internet surfers are using resolutions of 1024 x 768 or higher (see stats here, here, here and here).
However, as per the charts at the links above, a significant minority - 8% in the general browsing population - and perhaps higher in certain segments - are still using 800 x 600 screens. This situation is expected to persist for several years.
Whether a law firm designing a new website or blog needs to worry about "800x600" users depends on the clients you are targeting. If you cater almost exclusively to a tech savvy (generally B2B) audience likely using the latest technologies, then you can probably optimize for 1024 x 768, and not worry about 800 x 600. The benefit is that the extra 200 or so pixels of horizontal space allows you to display more content on your website in a bolder, more visually pleasing manner.
However, if you have a significant audience of lower tech users (e.g., consumer clients) possibly using 800x600 screens, then you should accommodate them.
Fortunately, you don't have to guess. An analytics program like Google Analytics can provide you with detail on the percentage of your visitors using higher resolution screens (1024 x 768 or higher) versus lower resolution screens (800 x 600 or lower).
As an example of how these considerations play out, see the following blog post in which the writer explains why he designed his own website (CSS Tricks) out to 980 pixels wide, but would redesign an e-commerce site selling baseball equipment to low-tech users to accommodate 800x600 screens (based on data obtained from Google Analytics showing that over 13% of visitors to that e-commerce site use 800 x 600 screens).
Here's a tool to determine the screen size for which any site is optimized (i.e., if you plug in a URL and then click 800 x 600, and the site gets cut off on the right, then the site is designed for a higher resolution).