Lysa Arryn was originally Lysa Tully and get this: she wasn't always crazy. She and Catelyn used to have fun as kids together with Petyr Baelish. But time has not been kind to Lysa, as Catelyn thinks when she sees her sister (35 Catelyn 6.110). Lysa has retreated to the Aerie, the castle of the Arryns, and refuses to help her sister. In fact, when Catelyn brings Tyrion to the Aerie, Lysa is furious at her for dragging her into this fight. Lysa foolishly lets Tyrion have a trial, and when he wins, Lysa has to let him go. Wah wah.
Lysa has had a lot of trouble getting (and staying) pregnant, and in the end, she only had one kid (compared to Catelyn's forty…yeah). Also, her marriage to Jon Arryn was a political act that never seemed to become the lovey-dovey marriage that Catelyn has (35 Catelyn 6.47). So Lysa is maybe a version of Catelyn that went down a sadder path. Or a crazier path. You decide.
Lysa Arryn doesn't have a lot in her life: her husband is dead; her young son is technically the Lord of the Vale, but a lot of men are trying to marry her in order to take over; and all Lysa wants is to be left out of the civil war between the Starks and the Lannisters. So can we blame her for spoiling her son, Robert?
Yes, yes, we can blame her. Robert joins Joffrey in the category of "crazy, cruel children." Robert is only six years old, but he already has a hobby: ordering people thrown out of the castle, that is, thrown to their deaths (35 Catelyn 6.39). He's also still breast-feeding (35 Catelyn 6.123). So, basically he hits all of the red flags for children: weird relationship with mom (check), sadistic tendencies (check), and… actually, that's all you need.
Robert also has some sort of epilepsy or "shaking sickness," which leads to the sad moment when Lysa repeats (and misunderstands) Jon Arryn's last words. He said, "Tell them the seed is strong," by which he meant that Robert Baratheon's seed was strong: that's why all his kids had black hair. But Lysa thinks Jon means that his own seed (ew) is strong and that Robert Arryn will survive and be well.
(Also, how messed up is it that a six-year-old can be the lord of a major part of the Seven Kingdoms? This society is a little kooky. Or maybe we're just jealous that we didn't get to be lords when we were six.)
Like all the great houses, House Arryn has many servants. As in other cases, Lady Lysa's servants help us see what kind of a person she is. For instance, in her court, Lysa keeps Maester Coleman, whose main job is to watch over sickly Lord Robert (which emphasizes how obsessed Lysa is with Robert); and Mord, the cruel jailor (which emphasizes how unheroic Lysa is: Catelyn Stark would never keep a cruel jailer, would she?).
And don't forget about Ser Vardis Egen, the captain of the guard. Vardis sticks out as the only knight at court who doesn't want to fight Tyrion. He seems to be the only knight who realizes how unfair it would be for a knight to slaughter a dwarf. As you might expect, since Vardis is a nice guy, Lysa Arryn orders him to fight; and Vardis doesn't survive long when Bronn is named as Tyrion's champion.
There's one other servant of particular note: Mya Stone. Mya helps Catelyn Stark up the mountain to the Aerie, guiding her over some steep cliff-paths. But here's why she's interesting to us: Mya Stone is an illegitimate child (the name "Stone" is the illegitimate child's name in the Vale, like "Snow" in the north) with black hair and a pretty fearless attitude. Does that remind you of anyone: say, Robert Baratheon? It's only later that Eddard comes out and says it: and that's how we know for sure that Mya Stone is one of Robert Baratheon's illegitimate children (48 Eddard 13.50).
House Arryn has many houses sworn to it, and we get a chance to meet some of these bannermen. Keep this in mind: many of them want to rule over the Vale, and some of them want to marry the widowed Lysa Arryn. (Because, of course, a woman can't rule by herself – don't be ridiculous. But do see "Gender" for more on that.) So there's kind of a weird dynamic going on here.
For instance, the Waynwood family, led by mother Anya, includes at least one person trying to marry Lysa, Morton Waynwood. (There's also Ser Donnel and Wallace Waynwood, but they don't seem to want to marry Lysa, they just hang around the court.) Other nobles who want this same thing include the old Eon Hunter and the young Lyn Corbray.
By contrast, Lord Nestor Royce thinks that he should rule over the Vale until Robert is old enough because he was the high steward for Jon Arryn. (Nestor also has a son named Albar, which will get confusing in just a minute because… Does the Royce name seem familiar? We've met other Royces: Yohn, Andar, and Robar took part in the Hand's tournament; while Waymar Royce was part of the Night's Watch. Confusing, right? Well, at least this reminds us that there aren't that many noble houses; and it helps to bring these chapters together. It's all taking place in the same world.)