Louis-Philippe and Napoleon III
Louis XVII was Louis XVI's little son who died (probably of tuberculosis) during the Revolution. Louis XVIII was Louis XVI's brother. But right away, Napoleon tried to get back into power. The British defeated Napoleon again at Waterloo. Now Britain was really mad at France and made sure France lost some of its land, and had to pay huge fines to Britain, Austria, Germany, Prussia, and Russia, and had to pay to have their armies supervising France for five years.
These other countries also didn't give Louis XVIII much power. Instead, they put in a Constitution, the way people had in England more than a hundred years earlier, or in America thirty years earlier. The French constitution, combined with Napoleon's Code, said that everybody was equal, that there should be freedom of the press and freedom of religion, and that the government couldn't take away people's property. But nobody enforced these rules, so they didn't work very well. Slowly poor people lost power, and rich people got more power back.
Louis XVIII died of gangrene in 1824, when he was 69 years old, and his younger brother Charles X became king. Charles tried to get back all the power his older brother had before the French Revolution. He gave a lot of money to rich families whose land had been given to poor people in the Revolution. Charles also sent the French navy to attack Algeria. People got mad, and the July Revolution of the people of Paris threw Charles out in 1830. Now they tried having Charles' second cousin, Louis-Philippe, as king. But Louis-Philippe also made the rich richer and the poor poorer. In 1847, an economic recession made things even worse, and in 1848, which was a year of revolutions all over Europe, Louis-Philippe got thrown out too.
After Louis-Philippe, the revolutionaries proclaimed a Second Republic, with the motto "Liberty, Equality, Community". (The First Republic was the one during the French Revolution.) They let all the people vote for a government, but to their surprise people voted for a conservative, traditional government, not for the revolution. They brought in Napoleon III, the nephew of Napoleon I, as their new leader, but after only three years Napoleon III declared himself Emperor instead.