In most French surnames, "de" does not indicate nobility and is just a part of the name. In a minority of names, however, it indicates noble origin. Originally, this kind of name indicated what fief the holder was associated with; for example, Godefroy de Bouillon was either lord of Bouillon or directly related to such a lord. If a nobleman had a title, for example baron de Montmorency, the holder's children would all be described as de Montmorency.
Later things began to get more complicated, as nobles accumulated titles. For example, a fictitious nobleman, Hugues de Montreuil, is given a duchy by the king, and becomes Hugues de Montreuil, duc de Ploërmel. His son would be known as Jean de Montreuil de Ploërmel, or more simply, Jean de Montreuil-Ploërmel. To complicate things further, Hugues would probably have other titles besides the duchy which his sons would use. If Hugues de Montrueil were duc de Ploërmel and also comte de Sainte-Cécile, then his eldest son Jean de Montreuil de Ploërmel would probably be known in society as M. de Sainte-Cécile.
Normally, a nobleman who had a family name "de" somewhere, and a title "de" somewhere, used the title socially, but signed documents with the family name. Thus, Roger de Rabutin, comte de Bussy, would have been addressed as "monsieur de Bussy," but would have signed "Roger de Rabutin." A woman was known by her husband's name or title, but legally speaking, kept her maiden name. These rules were nevertheless not universally followed. Al this is further complicated by which name these people are known to posterity; our comte de Bussy is known in the history of French literature as Bussy-Rabutin.
Later, having "de" in the name bespoke nobility, and many nobles and non-nobles added "de" before their surnames, even if their name was not associated with a fief.