The history of Ancien Régime France has had many regencies, most notably the regencies of Anne de Beaujeu (1483-1491), Catherine de Médicis (1560-1563), Marie de Médicis (1610-1617), and Anne d'Autriche (1643-1651), but when you say the la Régence, it always means the period from 1715-1723 when France was ruled by Philippe II, duc d'Orléans in the name of the child king Louis XV, his first cousin twice removed.
Preceeding regencies almost always showed a female relative of the king, usually his mother, exercising his authority during his absence or minority. As substitute monarchs, they legally possessed all the authority of the king himself. Toward the end of Louis XIV's life, he and those around him realized that a Regency can create an unstable situation. There were particular concerns about the continuing status of the king's bastards, the duc du Maine and the comte de Toulouse, who had been given special precedence and rights of succession against all precedent. So the king's will maintained Philippe d'Orléans as Regent, but attempted to dilute his power by providing for the duc du Maine to be in charge of the young Louis XV's person and household. All of this was swept aside by the Parlement de Paris soon after the king's death in 1715, and Orléans found himself the undisputed ruler of France.
Immediately after Louis XIV's death, the Court dissolved itself as the nobility fled Versailles for Paris. Orléans installed the new young king in the Louvre, with himself close at hand in the Palais Royal. The center of political gravity shifted back to Paris, where it has remained until the present day. This change was more than geographical; the monarch and his Court could never again dominate the intellectual life of France. The fashionable (aristocrats and others) shifted their lifestyle from haunting royal antichambers to attending salons and receiving guests in their cozier Parisian hôtels particuliers, which had effects on dress and interior design.
The Regency is often described as an important turning-point in French history, and can in some ways be seen as a capsule for the 18th century in France (except for the Revolutionary era). It can be said that the 18th Century began in 1715, not 1701 (or 1700, if you like). Salons, epistolary novels, Voltaire, the rococo, the philosophes, all these found their first untrammeled expression during the Regency. This was also the period of the Mississipi Company and Law's System. It's also seen as a period of high society's revolt against the staid court of Louis XIV and Mme de Maintenon; the career of libertines began, of which the duc de Richelieu is the most famous example. (Some of this may explain Louis XV's character.)
Many of the political crises that prepared the Revolution had their roots in the Regency. For example, the Regent restored the right of remonstrance the the Parlement of Paris, which had been denied them under Louis XIV. Later in the reign, the Parlements would take advantage of this right, and prove to be troublesome adversaries to the monarchy, culminating in the crisis at the end of Louis XV's reign.
The Regency formally came to an end in 1723 when Louis XV came of age (at 13 years old). The Regency was dissolved, and the Regent became prime minister until his death in the same year. Nevertheless, the Regency returned the monarchy to rule by ministers; to Orléans' Dubois succeeded Louis XV's duc de Bourbon and Fleury.
Regency also refers to a style of interior design, just as it does in England. With its curves and delicacy, it is nascent rococo or Louis XV, but it hasn't yet completely shaken the larger dimension of the Louis XIV style.