Hey, it's the walls of Qarth! From the outside, it's pretty much just a giant wall, surrounded by a giant desert. There are some soldiers outside and they're kind of Ancient-Greecey. They come out to meet Daenerys, who doesn't really think of her handful of riders as a Dothraki horde.
She starts to introduce herself with "My name is Daenerys," but she's interrupted by a fat, bald guy who finishes it for her: "Daenerys Stormborn of the House Targaryen." He claims to be a humble merchant. Also, he's heard of her, but we knew that from the part where he knew her name and family. His own name, he assures her, is too long and difficult for her to pronounce. He is simply a trader of spices. He's a fat, bald guy who claims to be a humble dealer in spices? Fine. I'll call him Kingpin. He and his twelve buddies over there are the Thirteen, charged with protecting Qarth, which is the greatest city ever. His specific claim is that it's "the greatest city that ever was or ever will be," which is quite a claim. Dany asks about the city and she pronounces it "Quarth." That bugs me, because it's a mispronunciation based on a spelling she's never seen. She's only heard it pronounced it "Karth," so how did she get to "Quarth"? The Kingpin would like to see her dragons. She's taken aback and doesn't want to show them off, possibly because they're very, very expensive. She would like to get inside and get some food and water before breaking out the CGI. Kingpin makes it clear that he personally doesn't think she was lying about having dragons, but other people are skeptical. You know how other people can be: Jerks, most of them. Daenerys gets her back up and announces, "Where I come from, guests are treated with respect, not insulted at the gates." Kingpin smiles the smile of someone who knows he's about to get off a great line and answers, "Then perhaps you should return to where you come from. We wish you well."
Daenerys says they promised to receive her. He feels that they've received her just fine. They just did so outside the gates. Then he gets down to what might be the actual issue: "Qarth did not become the greatest city that ever was or will be by letting Dothraki savages through its gates!" Technically, Dany isn't Dothraki, but I don't know the word for "a savage from Westeros." The Thirteen are going back inside their city. Dany tells them that when her dragons have grown, they'll lay waste to armies and burn cities to the ground. And if they turn her away, Qarth is first on the list. Kingpin isn't impressed. I am, though! Another of the Thirteen, whose name sounds like "Zaro-Zaron Duck-sauce" but is really "Xaro Xhoan Daxos," doesn't want to retreat. Kingpin is outraged. Daxos explains, "She is the mother of dragons. Do you expect her to see her children starve without breathing fire?" Kingpin says the decision is final. Daxos invokes Sumai, which is a vaguely defined Qarthian rule. He vouches for her, her people and the dragons. He cuts his hand and shows off his palm. Kingpin is unhappy: "Be it on your head!" But they get to go in. The gates are thrown open and it does look pretty nice in there.