What is an AngularJS Directive?
What it is (see the clear definition of jQuery as an example)?
A directive is essentially a function† that executes when the Angular compiler finds it in the DOM. The function(s) can do almost anything, which is why I think it is rather difficult to define what a directive is. Each directive has a name (like ng-repeat, tabs, make-up-your-own) and each directive determines where it can be used: element, attribute, class, in a comment.
† A directive normally only has a (post)link function. A complicated directive could have a compile function, a pre-link function, and a post-link function.
What practical problems and situations is it intended to address?
The most powerful thing directives can do is extend HTML. Your extensions are a Domain Specific Language (DSL) for building your application. E.g., if your application runs an online shopping site, you can extend HTML to have "shopping-cart", "coupon", "specials", etc. directives -- whatever words or objects or concepts are more natural to use within the "online shopping" domain, rather than "div"s and "span"s (as @WTK already mentioned).
Directives can also componentize HTML -- group a bunch of HTML into some reusable component. If you find yourself using ng-include to pull in lots of HTML, it is probably time to refactor into directives.
What design pattern does it embody, or alternatively, how does it fit into the purported MVC/MVW mission of angularjs
Directives are where you manipulate the DOM and catch DOM events. This is why the directive's compile and link functions both receive the "element" as an argument.
In HTML we have things like
<table><tr><th>. How would you describe what a, href, img, src, br, table, tr, and th are? That's what a directive is.