I've arbitrarily chosen three armchairs to illustrate the differences between the three styles. Keep in mind that styles are fluid, so not all chairs from each era have every characteristic described here. Also, between each stylistic period was a transition, so lots of 18th-century objects have characteristics that span a couple of these styles at a time.
The trajectory of furniture style in 18th-century France was, very simplified, from Baroque pomp and rigidity (Louis XIV) to sensuous Rococo intimacy and comfort (Louis XV) to austere Neo-Classical elegance (Louis XVI). You can see this in the armchairs, which start off quite throne-like and stolid, then become exuberantly playful and low to the ground, and then become restrained and delicately geometrical.
Obvious signifiers of style are curves vs. straight lines (S- and C-curves generally denote Louis XV or Rococo style), Classical motifs like fluting and acanthus leaves (Louis XVI or Neo-Classical style), and stretchers supporting the legs (Louis XIV or Baroque style). An interesting sign to look for is the arm rest, which got shorter during the Rococo era to accommodate the enormous panniers, or hoops, that women wore in their dresses at the time (the female silhouette underwent a similar stylistic trajectory, with their curves being most exaggerated during the Rococo era).
For more detailed explorations of these eras, read these posts from the Retrospect archive:
Sun King Style: The Decorative Arts of Louis XIV
Rococo Loco! The Style of Louis XV and Madame de Pompadour
Louis XVI Style: Revolutionary Road
Related post: Style Glossary: Queen Anne
Images: Louis XIV chair: France, early 18th century, at the Musée des Arts Décoratifs, Paris; Louis XV chair Nicolas-Quinibert Foliot, 1753, at the Metropolitan Museum of Art, New York; Louis XVI chair: Jean-Baptiste Bernard Demay, c. 1785-90, at the Musée des Arts Décoratifs in Paris.