King Robert and later King Joffrey Baratheon both have courts, which means that they have people around them (not "courts" in the sense of trials – though they have that, too). When you read "court" you can think "Hollywood star's entourage" since the courts are made up both of people who are there to help the king and people who just want things from him.
For an example of court, check out 17 Eddard 3 (from paragraph 9 on), where Robert holds an impromptu court to deal with Arya and Joffrey's fight; or check out 58 Sansa 5 (from paragraph 4 on) to see how many people attend Joffrey's first court (and how they pretty much all ignore Sansa since she's no longer cool). From those scenes, we can see what court is like and who is usually there: people dress nicely, it's mostly nobles, grand proclamations are read, and justice is dispensed. Or, in the case of Joffrey, random violence is dispensed (see 68 Sansa 6.38).
Maesters are people who have been trained in many scholarly areas, such as medicine and warfare (equally as cool, obviously). Each castle pretty much has its own master, so the Starks in Winterfell have Maester Luwin and the Arryns at the Aerie have Maester Coleman. In fact, as Grand Maester Pycelle explains, "My order [the maesters] serves the realm, not the ruler. Once I counseled King Aerys as loyally as I counsel King Robert now…" (34 Eddard 8.28). So, if anyone ever took over one of these castles, the maester would stay on to advise the new owner.
The royal family in the Red Keep (the castle in King's Landing) has the best maester of them all: the Grand Maester. Ooh. And they have him no matter who they are. Pycelle served the Targaryens and then the Baratheons; and if the Lannisters took over, he would serve them, too.
In fact, Pycelle already seems to serve the Lannisters, or at least Queen Cersei. Although Pycelle has the title of "Grand Maester," he doesn't seem so great to us, mostly because he's totally a toady for Queen Cersei. When Eddard tells Pycelle something, he's sure that the queen will hear it soon (46 Eddard 12.11). Basically, this guy is a good reminder that not everyone at the court has Robert's best interests in mind.
Ser Ilyn Payne is the royal executioner and proof that our author isn't above making a silly joke out of a name: "Payne" sounds like "pain." But though he has a jokey name, he's also presented to us as kind of a dangerous, almost monstrous guy: he has no tongue, so he can't speak, but he also never smiles, and just looks vaguely skeletal all day. At least, that's what Sansa thinks whenever she sees him: "He made her feel as though something dead were slithering over her naked skin" (45 Sansa 3.12). Ick.
Moon Boy is the court jester/fool, something that every court needs. Instead of watching cats playing Patty-cake over on YouTube, medieval kings might watch their jester do similarly silly things. One of the great things about being the jester is that you can say stuff that no one else can say, mostly because people just assume you're stupid. But as Sansa notes, Moon Boy's jokes are so sharp that he might not be so dim after all (30 Sansa 2.28). We kind of wonder what his version of this book would be.
The High Septon is something like the Pope: he's the leader of the Faith. Since this book isn't full of religious arguments (although there are some), he doesn't play too big a role. The High Septon is there when Eddard "confesses" and is executed (no quotation marks around that) (66 Arya 5), but we don't see a lot of him. In fact, he's not really part of the Baratheon court, but since he's a powerful person, we put him here.