Paris began on an island in the Seine, the Île de la Cité, around the third century BC, when it was called Lutetia, and you can still stand on that island and see most of what came after: the cathedral begun in 1163, the king's chapel with walls of stained glass, the oldest standing bridge, a fortress that became a palace and then a museum, and an iron tower that was meant to come down after the 1889 World's Fair. No other European capital has so many places that everyone has already heard of. That is exactly the problem.
The hard part of Paris is not choosing what to see. It's that the famous places run on timed tickets, closing days and queues, and a day planned badly turns into two hours in a line for a ticket you could have booked from your hotel. This plan is built around that. Every day sticks to one part of the city, because the real cost in Paris is moving between sights, and no big museum falls on its closing day. Five days is the number that works: four for the center, one for Versailles. It's a full plan, and day two is the heaviest.
Day one is the Île de la Cité and the Latin Quarter, day two the Louvre, the Marais and the Tuileries, day three the Left Bank from the Musée d'Orsay to the Eiffel Tower, day four Montmartre and the Arc de Triomphe, and day five Versailles. Start on a Tuesday if you can. The Louvre and the Orangerie are closed on Tuesdays, Orsay and Versailles on Mondays, and Tuesday to Saturday dodges every one of them. If you land on a Monday, do day four first, since Montmartre has no closing day, then days one, two, three and five. For the bigger picture of the country, see France.