The translator of this novel says in his introduction that "no story in the world is more exciting than The Black Sheep, combining as it does the compelling readability of the blood-and-thunder with the deeper insights of literary art." Well, he's wrong, and this isn't even the most exciting Balzac I've read, but it does have its moments, and the pace increases remarkably up to the climax.
As ever with Balzac, the plot is mainly concerned with money, in this case an inheritance. The book starts with a tortuous introduction of the dramatis personae, being: Jean-Jacques Rouget, who has inherited the bulk of his father's wealth, due to the latter not believing that Agathe, Rouget's sister, was his daughter. Agathe, a beautiful and pious woman, has two sons, Joseph, studious, kind, and a very talented painter, and Philippe, a swashbuckling soldier, and a scoundrel.
Agathe's husband was a chief civil servant of Napoleon's state, but once he dies, and the monarchy is restored, she loses that income, and is dependent upon the income from the modest investments she has. Philippe, who she adores, deceives her and gambles away her capital, so that she's forced to try to appeal to her brother, and ensure that she has a share of his legacy. Jean-Jacques, however, is weak-willed, and controlled by his mistress, Flore, with whom he's obsessed. She in her turn is in love with a local ex-soldier and leader of a gang of pranksters, Maxence Gilet.
So far, so complicated. Add to this a political conspiracy, for which Philippe is convicted, then a confrontation and duel between Philippe and Maxence, and the full intricate skill of Balzac's plotting is evident. The book is uneven though, with digressions in which Balzac seems keen to exhibit his research. You can feel the energy with which Balzac wrote it, fuelled on caffeine, but slightly undisciplined. There's also a little too much narratorial judgement on the characters, less than in, say, Hugo or Dumas, but far more than in one of his later masterpieces such as Cousine Bette.
The themes are of money and status, but also of legitimacy. There are several characters whose fortune depends upon their legitimacy being recognised - Agathe is denied her full inheritance as her own father denies his inheritance, Maxence is presumed by many to be Jean-Jacques' brother, and Flore is a mistress rather than a wife. The doubt about the status of these characters plainly parallels the political turmoil running behind the main plot - the legitimacy of the Bourbon throne, and the resistance to it of both Philippe and Max. Much of the political significance of the novel will pass modern, and particularly English, readers by, but it would have been at the forefront of contemporary readers' minds, during the Orleanist rule.
Further themes, of the corrupting power of money, are typical of Balzac, and the ending, in which the bad guy gets ruined and the good guy gets a fortune, is rushed and unconvincing. There's a sense that, once Balzac had passed the climax of the duel, which is dealt with briefly, he had little energy left to sustain the novel.