The Musketeers of the Guard (Mousquetaires de la garde) were originally founded by Louis XIII. in 1622 when he equipped a company of Chevaulegers, who were primarily light lancers, with the new muskets. They served as royal guards, so their main responsibility was the protection of the king as part of the King's Household Army. The name "musketeers" is a bit confusing since it traditionally refers to musket-wielding infantry and the Musketeers of the Guard were a mounted regiment, so actually more like dragoons.

Shortly after a second company of Musketeers was founded as the personal guard for Cardinal Richelieu, dressed in the church's red. The Cardinal's absolutistic reforms, which attempted to shift power from the nobility to the king himself, were understandably much hated amongst the nobles. Since the Musketeers recruited solely from nobility that must have been a bit unfortunate for the Cardinal, the commander of the king's own Musketeers, the Comte de Tréville, even took part in one of the many plots against Richelieu.

Despite all that the Musketeers actually survived the Cardinal and the king's company was only disbanded by his successor Cardinal Mazarin in 1646, four years after Richelieu's death, for being "too turbulent", only to be reinstated eleven years later. After Mazarin's death in 1661 his own company came under command of the king as well, who subjected both of them now to a big reform.

His own company became the "Black Musketeers" (mousquetaires noirs) and took the motto Alterius Jovis altera tela, while the former Cardinal's company became the "Grey Musketeers" (mousquetaires gris) with the motto Quo ruit et lethum; both named after the colours of their horses. Unfortunately, my Latin and French are equally terrible by disuse so I can't give you the English mottos.
The size of the companies is also increased due to popular demand, since it became a requirement for officers to have prior military experience and many preferred to get that in the privileged Household Army.

Every member of the Musketeers had to dress and equip themselves at their own expense, including horse and sword since the king only provided the muskets. Out of that, the famous blue cape (if that is the right word) that every single film uses arose in an attempt to save them money, by making that the only uniform to be worn over whatever the soldiers fancied. But because that thing was incredibly annoying in actual combat the uniforms underwent some drastic changes over the next decades, here is an overview.

In 1776 the Musketeers were disbanded due to financial reasons, then reformed by the king in 1789 only to be abolished by the new French Republic right away in 1792. After the return of the Bourbons both companies were once again recreated, then disbanded under Napoleon's Hundred Days, then recreated after his final defeat. But because the royal guard of the Restoration wasn't the glorious position it once was only a small part of the new Musketeers had actual experience and it certainly wasn't worthy of its elite status anymore. So it was finally and officially ended on the 31^st of December 1815.