Web designers have interesting territory in which to work. On the one hand there is the constantly evolving landscape of HTML and CSS standards, where developers are keen to be using the latest and coolest technologies. This keeps developers interested in what they are doing. On the other hand there is a need to write websites that look "right" on the end-user's system: more of an artistic, design parameter than a technical one, and the thing that excites graphic designers and marketing people.
The World Wide Web was adopted by marketing departments in the middle of the 1990s as they realised that this was a cool alternative to printed advertising. Ever since then, Marketing has been responsible for keeping the money flowing in web development. The appearance of the site is a high priority, and the customer is always right - even when what [s]he wants is technically difficult.
The "Browser War" is about the incompatibility between the main browsers : mostly IE6, IE7 and FireFox at the moment: the same piece of HTML may display slightly differently on each. This is not acceptable to designers. The web is full of articles describing how to get around the quirks of the different browsers: how to write sites that use semi-transparent PNG files (and degrade nicely to IE6 which doesn't support them); how to cope with text resizing (IE6 only resizes text; IE7 resizes everything except the body background image, and both IE7 and FireFox let you resize either everything or just the text).
How do we know that IE6 is still important, and what about the other browsers? The web-advert host companies track browser usage from billions of visits, to get a reasonable picture of the browser and OS landscape. There's a helpful Wikipedia page that tracks the numbers and explains much about how to interpret the statistics. Interesting facts are that Firefox is more popular among technical people (hence the higher Firefox scores at W3Counter, w3schools and the like), and that Opera has a very large following in Russia (28% at mail.ru, and as high as 36% according to StatCounter).
Digging into the IE share by version number we see that IE6 still has around 25% world usage. This is apparently because it is the default browser in Windows XP. With an audience of billions, the web design team can't ignore a platform used by so many. In my own work, I have a simple bit of server-side code that detects certain browsers and can deliver alternative content for IE6. For the rest, I stick to the standards, because it's a simple way of reaching the rest of the browsers without additional coding. Thus I generally don't have to worry about supporting Chrome, Opera, Konqueror, or Safari. By sticking to the standards and testing the page with Styles switched off, I'm also confident that my site is usable by those with vision impairment. Some estimates rate this category at around 10% of browser users, so it's clearly important.
Faced with the problem of coding for different browsers, it is hardly surprising that some web developers adopt Adobe Flash for entire websites. The graphics are highly controlable, and the end user experience is guaranteed to be the same across (almost) all browsers. But the end result takes longer to load, isn't accessible to audio browsers (screen readers) or those who don't (can't) install Flash, (which most important from a marketing perspective isn't supported on the iPhone).
The landscape of browser usage is changing in a dramatic way. More and more people use the Internet on their smartphones whilst out and about. Steve Jobs claimed in 2008 that the iPhone accounts for 71% of mobile browser usage in the USA (the iPhone uses the Safari browser). Opera also seems to be popular on Mobiles. There are over 4 billion mobile phones in use around the world. The mobile world presents additional design problems - mostly the small screen space. Web developers especially need to address mobile in order to reach their customers.
The most popular browsers today appear at first glance to be IE6, IE7 and Firefox. But taking into consideration the Russians, the blind, users of mobiles and everyone else, it becomes clear that the Browser War is far from over. In the non-desktop space, Microsoft is a long way behind, but there is no one dominant player. The need to adhere to standards and to test everywhere is greater than ever.