I know what makes an IPA an IPA, but what are the unique characteristics of it's common variants? To be specific, the ones I'm interested in are Double IPA and Black IPA, but general differences between any other styles would be welcome too.
This is a broad question, notably regarding the differences between IPAs and Double IPAs, but here's an overview focusing on the naming (and misconceptions thereof) and brewing of variants.
Double IPA vs. Imperial IPA (IIPA)
First, the Double IPA is also known as an Imperial IPA (IIPA). You can think of the "double" as referring to the two letter I's :-) but the name actually owes itself to the beer having "double" the strength / hops.
IPA vs. Double IPA
The Double IPA generally uses more hops (though not literally "double"), lengthier brewing processes (e.g. Dogfish Head's 90 Minute IPA, which is boiled for longer than its 60 Minute variant), sometimes secondary fermentation, etc. It has higher alcohol content, IPAs stopping around 7.5% ABV, while Double IPAs can soar to above 10%.
Double IPAs are claimed to have been started by the owner of the Russian River Brewing Company, famed for its flagship brew, Pliny the Elder, a Double IPA itself.
American Pale Ale (APA) vs. American IPA
The APA and American IPA are not synonymous. American Pale Ales are a derivative of (regular) British Pale Ales, deriving their own flavor from American hops, e.g. Sierra Nevada's Pale Ale. American IPAs, meanwhile, are, of course, derived from actual (British) IPAs, and possess greater bitterness and higher alcohol content—examples include Sierra Nevada's Celebration Ale and Dogfish Head's 60 Minute IPA. The line between APAs and IPAs blur (as do between many variants of ales), but categorically speaking, APAs are not IPAs.
IPA vs. Black IPA
The black IPA (also known by American Black Ale—and yes, a subset of the American IPA) uses dark roasted malts (used in porters and stouts) to achieve the color, while preserving hops characteristics. Some consider the black IPA more of a hoppy porter instead—and don't believe the style will be formally recognized (e.g. by the Brewers Association).