Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Third Person (Limited Omniscient)
For this novel, Henry James goes with a third-person limited omniscient narrator, meaning that he's talking in the third person, but he limits us to knowing only what Strether knows. And that works well for the purposes of this book, which is all about the difference between what Strether knows and what the people around him do (which is, you guessed it, a bunch more).
In fact, you could argue that this book wouldn't have any suspense at all if James didn't use third-person limited omniscient. If we knew right away what the other characters did, we'd know from the start what Sarah Pocock is thinking during her meetings with Strether, and it wouldn't take us 50 pages to finally figure it out.
Further, we'd know what everyone except Sterther seems to know, which is that Chad and Madame de Vionnet are having an affair. That little nugget of information takes us 500 pages to figure out, and when it finally pops into the open, Strether's whole world comes crashing down.
One interesting thing about James' narrative voice, though, is that sometimes the omniscient narrator will peek out from behind James' language and say something like, "All sorts of things in fact now seemed to come over him, comparatively few of which his chronicler can hope for space to mention" (2.1.4). The narrator, in this case, actually identifies himself (or herself) as Strether's "chronicler." This is an isolated and rare event in the text, but it has the strange effect of making us think that this book's narrator is sitting at a desk somewhere and writing down Strether's life. Which, um, is exactly what James was doing.
The greatest part about going with an omniscient narrator is that is also allows Henry James to slap down some amazing descriptions of Paris and its inhabitants. If he had gone with just a first-person narrator, we'd be stuck with whatever descriptions Strether could believably come up with himself. It'd still be good, but not Henry James good.