Petyr Baelish is King Robert Baratheon's master of coin. Petyr was actually fostered with the Tullys, so he grew up with Catelyn, Lysa, and Edmure. He apparently had a crush on one or both Tully sisters, which complicates their interactions when they meet later in life. Baelish is in charge of taxes and payments for the royal government, and he retains his position even after Cersei takes over. (Maybe he holds on to that position, in part, because of his bribery of Janos Slynt, captain of the City Watch.)
The Fantastic Plastic Petyr
Here's the basic fact about Petyr "Littlefinger" Baelish: we don't know whose side he is really on. Varys calls him "the second most devious man in the Seven Kingdoms" (59 Eddard 15.65) – but why should we trust Varys, the first mostdevious man? So any discussion of Petyr is going to involve a lot of "seems"s and "supposedly"s and "liar, liar, pants on fire"s.
For instance, Catelyn remembers that, as a boy, Petyr could always get into trouble and then seem to be sorry about it; and now that she sees him again she sees that "The years had not changed him much" (19 Catelyn 4.58). So, basically anything Petyr says may be a lie. As Eddard realizes about him, Baelish loves intrigue (21 Eddard 4.56).
More examples! Petyr doesn't object to assassinating Daenerys, unlike Eddard. But when Petyr talks to Eddard later, he says that he helped Daenerys more than Eddard did: "If truth be told, I did the Targaryen girl more good than you with all your talk of honor" (34 Eddard 8.69). How? By staying in the discussion and leading them to take a cheap way to assassination? So, does that mean that he actually doesn't want Daenerys executed? Or does he just want to save money? Or does he change his wants according to the situation? Yeah, that last one sounds about right.
It's hard to know where Petyr stands or what he stands for (sometimes because of his sarcastic, dry sense of humor). We don't want to take Varys' word for it, but Varys notes, "Littlefinger loves Littlefinger" (31 Eddard 7.141). And that seems pretty accurate. So maybe all of this is just a way for him to make himself more powerful and important. Or simply a way to let himself survive. (After all, it's not like Petyr wants to be king, right? Right?)
About that Name…
In case anyone asks, "Littlefinger" is his nickname because (a) his family is from the smallest of the Finger islands; (b) he was always small, even as a child; and (c) Edmure Tully decided to call him that. Does it seem like a mean nickname? Do you think Littlefinger resents the way that the Tully family treated him? Oh, great, now we have revenge as a possible motive.
Minor Characters Connected to Petyr
Janos Slynt is the "frog-faced" captain of the City Watch who sells his loyalty to the highest bidder (55 Sansa 5.18). And what's worse: the highest bidder is the Lannister group. That should cause us to pause: we might be uncomfortable with a police officer (Janos) taking a bribe. But if he had only taken the bribe from the right person (Eddard), would we still be uncomfortable with it? Now who's the bad guy? (Still not Shmoop, of course.)
Maybe, in fact, Janos sides with the Lannisters because they'll raise him to the title of lord and that will help out his three kids (Morros, Jothos, and Danos). Maybe "frog-faced" Janos is just trying to do the best thing for his family. Ugh, don't you hate it when people do the wrong thing for reasons that seem reasonable? It's hard to tell what Janos' real motivation is, but it sure makes us feel uncomfortable that Eddard Stark tried (and failed) to bribe him.