Like many people, you probably get in your car a number of times a week and drive somewhere, whether it be to work, the supermarket, or to take the kids to school.
In doing so, you follow a number of conventions. You stop at red lights, you drive on the correct side of the road for your country, and you turn your lights on at night so that other people can see you.
You also assume that other people will do the same, otherwise using roads would be an extremely dangerous activity and you could never be sure whether you would get where you were going alive.
Web standards are not a matter of life and death like road rules, however they do make life a lot simpler.
A web site that is designed to be standards compliant should be viewable on any computer with a standards compliant web browser.
Unfortunately for a long time, Microsoft has dominated the web browser market with Internet Explorer and older versions are particularly buggy and non standards compliant. What this means is a site that looks great on old versions of Internet Explorer will quite likely look horrible on other web browsers including newer versons of Internet Explorer.
If Internet Explorer is so popular, why worry about alternatives? Plenty of people use Macintosh computers, and they use Safari as their default web browser, also the Nintendo Wii games console is able to surf the web with Opera, and then of course there's the popular Firefox web browser which many people have adopted as a generally more secure alternative to Internet Explorer.
Being able to design a web page once, according to standards and know that it will display as intended is a huge cost saving, compared to designing for any and every quirk of non standards compliant browsers.
If you're using and old version of Internet Explorer (version 6.0 or below) or any other old browser you can do yourself and web designers a big favour by upgrading to one of the new more standards compliant browsers. The good news is they're all free downloads, and apart from Internet Explorer work on most operating systems including Windows, Mac and Linux.