"Good meals at moderate prices". This subset of the Michelin Red Guide is titled Les Bonnes Petites Tables du Guide Michelin. The restaurants listed here may or may not also have a Michelin Star or two. But where Michelin Star restaurants are about quality and glamour and forget the price, the Bib Gourmand restaurants are about a good blend of quality-and-price.
A Bib Gourmand restaurant must offer a quality menu for 29 € or less (35 € or less in Paris). The 2010 version is subtitled: The 555 best restaurants at less that 29€, with over 100 new places..
The low-cost menu on offer isn't necessarily the limit at these restaurants. For example, Oscar's in Antibes has menus from 28 - 56€.
The Bib Gourmand restaurants were introduced in 1997. The Bib Gourmand's Les Bonnes Petites Tables du Guide Michelin is published every year on the first of February. "Bib Gourmand" is a classification given to restaurants that meet Michelin's requirments for this category.
All of the Bib Gourmand restaurants are listed in the standard Red Guide (Guide Rouge). In addition, a separate Bib Gourmand Guide (Les Bonnes Petites Tables du Guide Michelin) is available that lists only the restaurants with this Bib rating.
The name "Bib" is not a reflection on needing to protect your front from messy eating, but "Bibendum", Michelin's Michelin-Man logo.
The Michelin website (ViaMichelin) says the Bib Gourmand rating began in 1997. Wikipedia (English) gives the date as 1955.