Consider your average american IPA. What food would go well with it?
As a complete novice to food pairings with beer, I've been looking at this nifty pairing chart on http://www.craftbeer.com [PDF] whenever pairing questions have come up on Beer.SE. According to it, the suggested pairing for an IPA is
Strong, spicy food (classic with curry!); bold, sweet desserts like carrot cake
and for a Double/Imperial IPA is
Smoked beef brisket, grilled lamb; Southern chicken-fried steak.
I found a great little summary by Michael Agnew (2012) that explains some of the rationale behind these pairings (emphasis mine hereon):
When pairing IPA with food you have three basic flavor hooks at your disposal; bitterness, hop flavor (spicy, grassy, herbal, earthy, and citrus), and caramel. Hop flavors have a great affinity for spices and light fruits. Bitterness has a cooling affect. Paired with spicy dishes, IPA will fan the flames at first, but douse them in the end. Bitterness also amplifies salty and umami flavors. The caramel flavors in the beer will latch onto the sweeter side of a dish, tying into things like caramelized onion or the crispy skins of roast poultry. And the hop acids and carbonation make IPAs great palate cleansers to take on even the fattiest deep-fried delights.
Samuel Adams has a "Pair with Sam" tool, and looking at its suggested pairings for their Whitewater IPA (hoppier than their Latitude 48 IPA), pork appears to be the meat of choice: pulled pork, barbeque ribs, chorizo, roasted pork tenderloin, and sweet sausage; and the rest of its suggestions align with the salty-and-spicy pattern we've seen.
We seem to have found strong consensus already, but just one more to make sure. In Blue Point Brewing Company's words, its IPA pairings are meant to complement
intensely flavorful, highly spiced dishes, such as curry, and bold, sweet desserts like flourless chocolate cake and crème brulée
rich, aromatic, spicy and smoked foods such as chili, BBQ ribs, grilled chicken, and beef.
Actually, I think they stole those words from Wegmans' Guide to Beer and Food Pairing.
CHEESE! Extremely sharp cheddar is best. A sharp Vermont or New York white cheddar will do as well.
My personal favorite is Tillamook Special Reserve Extra Sharp.
Cured meats - the saltiness of the meat and bitterness in the IPA play well together.
Be mindful of the variety of IPA on the US market -- the average New England IPA is a different beast than the average west coast IPA. I'll call out a few excerpts from the serious eats article Andrew Cheong quotes above:
IPAs that lean extra-heavily on bitterness can be a bit tricky to match with food, which can make the beer seem astringent.
...IPAs that emphasize hop flavor and aroma over bitterness...I especially like these beers when served with Indian food. Hop flavor melds wonderfully with common Indian spices like tamarind, coriander, and cardamom.
Mexican food mixes light and dark flavors like cilantro and refried beans, lime and roasted chilies. Those combinations make great partners for IPA with its own caramel/citrus combo. Stick with lighter-bodied beers here, since bigger brews can easily overwhelm.
My personal experience -- I predominantly drink IPA these days -- is that I haven't had a memorably bad pairing, save for once where an overly sweet cupcake clashed with the bitterness of the IPA. But that was likely an exceptionally high-IBU beer. (I suddenly feel the need to gather more data).
Spicy foods. My personal favorite is stir fry. Something that isn't heavy on carbs is ideal, so with stir fry I go light on the noodles (or rice) and heavy on peppers, onions, and meat (usually chicken or beef). Again, don't use the milder sauces like teriyaki, use something with some kick.