If you're not familiar with Richard Nixon, don't worry about it. Neither does the show. This story has all the historical acumen and insight we've come to expect in the last year or so:
Doctor: "Vietnam, Watergate..."
River: "There's some good stuff, too..."
Doctor: "Not enough."
So the secret thing that Canton needs to investigate is how the President keeps getting mysterious phone calls from a mysterious child, even though there's no way for him to get direct calls in the Oval Office. So what you have is not only a mysterious spacesuit like in "Silence In The Library," but also mysterious spooky imperiled voices, like in... Oh, every Moffat episode, really. (But if you're thinking the "Silence" in the Library has anything to do with the "Silence" of this and last season, think again. That would imply a plan of some kind, and not just a dartboard full of cool shit.)
Because women can't drive, it's hilarious and faux-empowering that River Song can not only hush the TARDIS engines but put "the outer shell on Invisible," because you'd expect that she wouldn't be able to work the TARDIS but in fact she works it better than the male that usually operates it, so you see that's a subversion of expectation which is after all the root of most comedy. Women, am I right?
"You can turn the TARDIS invisible?" the Ponds ask wonderingly, having missed out -- much like the script itself, it seems -- on the whole thing about how TARDISes can look like anything whatsoever when they're working right. More made-up details about how the TARDIS works when she's "cloaked," while in the Oval Office Nixon plays a recording for Canton:
Mr. President? I'm scared, Mr. President. I'm scared of the Spaceman..." They talk about how it's not a little girl, but a little boy -- another strange detail we'll probably but not necessarily be revisiting, since it's a little girl in the end -- and the kid says her name is Jefferson Adams Hamilton before hanging up. Nixon explains that he's come to Canton because he gets the calls absolutely anywhere he goes, and that he can't trust the FBI.
(Running with the sub-elementary school understanding of history in which this show seems to revel, Nixon's paranoia and tendency to record conversations are mentioned hilariously in every scene, not unlike a cruddy Saturday Night Live sketch about a famous historical figure. Ditto the same joke, approximately twelve times in this episode, about how Americans shoot things all the time. With those guns we love so much.)