You can read Greater Hippias online in English at the Perseus Project or in French at Wikisource. Nickolas Pappas's SEP article on Plato's Aesthetics has a short but handy summary of some of the most important ideas of the dialogue. But you can get an even shorter handy summary from the last line of the dialogue, which gives one of the few things Plato's Socrates ever claims to know: Chalepa ta kala, the splendid things are difficult.
The dialogue is an apparently private discussion between Socrates and Hippias of Elias, the same polymathic sophist who is a character in Hippias Minor. It happens to be important for a key passage in the dialogue that Socrates is the only son of Sophroniscus.
The Plot and The Thought
Socrates opens the dialogue by hailing Hippias as splendid and wise, and notes that he hasn't been in Athens in a while. Hippias replies that he constantly gets sent on embassies for Elis, and has recently most often been sent to Sparta. Socrates remarks ironically that this is what it is to be wise: make a lot of money off of young people while providing very public services to the city. But, he notes, it's odd that the wise of antiquity, the Seven Sages of Greece, and some more recent wise men like Anaxagoras, have avoided such state affairs. Hippias replies that this was surely due to lack of ability, and agrees with Socrates' suggestion that the skill of sophists has progressed like the skills of others, although he says he always plays up the skill of the ancients on the principle that one should avoid the envy of the living and the wrath of the dead. Socrates remarks that Hippias is putting splendid thoughts into splendid words, and adds the examples of Gorgias and Prodicus and, earlier than they, Protagoras. Hippias replies that Socrates has no idea just how splendid it is, since he makes an immense amount of money. Socrates' response to this, of course, is highly ironic.
Socrates then asks if Sparta is where Hippias has made the most money, and this leads to a discussion of Sparta, the implication of which is that Hippias perhaps does not actually know how to make people virtuous, at least in the judgment of Sparta, which has the reputation for producing the most virtuous Greeks. Hippias relates that recently in Sparta he gave a speech in the person of Nestor, the famed wise man from the Trojan War, speaking to Neoptolemus, son of Achilles, on what sort of activities are splendid, i.e., "the sort of activities that would make someone famous if he adopted them while young" (286b). He is going to demonstrate the same speech at the invitation of Eudicus (who is mentioned in Hippias Minor, as well), and invites Socrates along. Socrates says he will "if all goes well" (286c), but says that Hippias has reminded him of someone who was insultingly questioning him about whether he would be able to say what the splendid was, and how he was completely stuck in trying to answer it.
On the basis of this Socrates gets Hippias to try to give a definition of the splendid, but Hippias turns out to have difficulty distinguishing between giving a definition of the splendid and giving examples everyone recognizes as splendid. (It is the kind of weakness to which polymaths are subject, in my experience.) His attempts at answering the question are:
(1) A splendid woman is a splendid thing.
(2) The splendid is gold.
(3) The most splendid is to achieve old age, to give your parents a splendid funeral, and to have a splendid funeral given by your children.
Socrates, of course, is able to show a number of problems with these suggestions. He then goes through some possibilities of his own (while playing off answers Hippias gives to the questions):
(1) The splendid is the appropriate (to prepon).
(2) The splendid is the useful (to chresimon).
(3) The splendid is the beneficial (ton ophelimon).
(4) The splendid is the kind of pleasant that comes to be through hearing or seeing (to meros tou hedeos to epi te opsei kai te akoe gignomenon).
Each of these has problems, though. The first seems to founder because it cannot be used to distinguish what is really splendid from what only appears to be so; the second runs into problems because usefulness is based on power, which can be used to do the base as well as the splendid; the third seems to make the splendid different from the good; and the fourth seems not actually to identify what the splendid is, since it does not say what it is that the pleasant through hearing and the pleasant through sight have in common.
Hippias gives his last account of what is splendid (or fine):
But here's what is fine and worth a lot: to be able to present a speech well and finely, in court or council or any other authority to whom you give the speech, to convince them and go home carrying not the smallest but the greatest of prizes, the successful defense of yourself, your property, and friends. (304b)
Hippias continues that Socrates' friend who keeps asking the annoying questions should abandon his quibbling. (Compare Callicles in Gorgias.) Socrates replies that he seems to be in a bad place; if he tries to compete with wise men like Hippias, he gets "mud-spattered" (304c) by their speeches; but if he agrees with them, then the annoying interlocutor insults him for not knowing what the splendid is. But, he says, it is probably for the best; as the proverb says, what's splendid is difficult. And thus the dialogue ends.
* Catherine Zuckert in Plato's Philosophers has an excellent summary of the implications of Hippias's proposals for the splendid/fine/beautiful/noble:
Readers get a sense of the customary noble pursuits Hippias would say that Nestor recommends to young men like Neoptolemos from the sophist's three attempts to say what is noble (or beautiful)--a noble virgin (whom a young man should take as wife), gold (which he should amass not merely to support but to adorn himself and his family), and finally a beautiful burial by one's own offspring (after having become rich, healthy, and honorable enough to reach old age and have provided a noble funeral for one's parents). The definitions correspond, loosely, to three stages in a man's adult life. (pp. 261-262)
This no doubt relates to Hippias's conception of what an irrefutable account is: saying what everyone thinks (288a); these answers are calculated to be the kinds of things that young men would have assumed without thinking constituted the successful life.
* As always with the aporetic dialogues, even though we never get an account of to kalon, the journey is not fruitless. We learn just how expansive the concept is, and that it has some connection to the good (ton agathon) and to pleasure. We also, and perhaps more importantly, have put into question what many of the ancient Greeks would have thought of as a splendid life.
Quotations are from Paul Woodruff's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 898-921.