What does !important mean in CSS?

It means, essentially, what it says; that 'this is important, ignore subsequent rules, and any usual specificity issues, apply this rule!'

In normal use a rule defined in an external stylesheet is overruled by a style defined in the head of the document, which, in turn, is overruled by an in-line style within the element itself (assuming equal specificity of the selectors). Defining a rule with the !important 'attribute' (?) discards the normal concerns as regards the 'later' rule overriding the 'earlier' ones.

Also, ordinarily, a more specific rule will override a less-specific rule. So:

a {
    /* css */
}

Is normally overruled by:

body div #elementID ul li a {
    /* css */
}

As the latter selector is more specific (and it doesn't, normally, matter where the more-specific selector is found (in the head or the external stylesheet) it will still override the less-specific selector (in-line style attributes will always override the 'more-', or the 'less-', specific selector as it's always more specific.

If, however, you add !important to the less-specific selector's CSS declaration, it will have priority.

Using !important has its purposes (though I struggle to think of them), but it's much like using a nuclear explosion to stop the foxes killing your chickens; yes, the foxes will be killed, but so will the chickens. And the neighbourhood.

It also makes debugging your CSS a nightmare (from personal, empirical, experience).

The !important rule is a way to make your CSS cascade but also have the rules you feel are most crucial always be applied. A rule that has the !important property will always be applied no matter where that rule appears in the CSS document.

So, if you have the following:

.class {
   color: red !important;
}
.outerClass .class {
   color: blue;
}

the rule with the important will be the one applied (not counting specificity)

I believe !important appeared in CSS1 so every browser supports it (IE4 to IE6 with a partial implementation, IE7+ full)

Also, it's something that you don't want to use pretty often, because if you're working with other people you can override other properties.