Classically in the world of web design and development you would write up the code for a site to be viewed in one of the few browsers such as netscape, internet explorer, opera, etc.... These browsers mostly all supported the same web standards and it wasn't too hard to make a site work across all browsers. That doesn't work out anymore.
In modern times we have hundreds of browsers on hundreds of platforms and they are all being viewed in one way or another. Some support advanced features and some don't. Because of this us designers and developers have to keep several things in mind and make sure that we're targeting our specific audience and making sure that they experience the best browsing experience possible.
For a long time internet explorer was the enemy and in some ways this still holds true. Overall Microsoft has worked hard to rid themselves of the horrible reputation that the browser has got. IE9 should tackle a lot of the issues of the past but unfortunately this doesn't mean people will upgrade.
Very similarly to the IE upgrading issue is the legacy phone issue. These days more and more people are moving from a computer to a phone for the basic computing needs such as email, web browsing, social networking, and even document management. In today's market it can be difficult to distinguish which device a person is using so it's very important to make your site very scalable. This means that a mobile site developed by myself should work not only on an old Blackberry Curve but also on an ipad without looking too funky. This is an issue considering current versions of iOS and Honeycomb support some of the most up to date standards including css3, html5, javascript, jquery, etc. and legacy models of phones do not. In fact, as I was developing a site the other day I found that older blackberrys do not support css float and margins. This is a huge deal when making a site considering that these are common practice. Contemporary designers and developers need to find ways to work around this by utilizing old but still relevant code to accomplish the same layout. Easier said than done ;)