Let's start with the basics: the concept of "lodges" initially comes from Freemasonry. Craft masonry, that is the ordinary lodges that confer the first three degrees (Entered Apprentice, Fellowcraft, Master Mason), typically uses blue regalia and were referred to informally as blue lodges. However, freemasonry really took off during the 18th, 19th and 20th century, and there developed several lodges with alternative rites - the Royal Arch Masonry, for example, confers four additional degrees about the initial three (Mark Master Mason, Past Master, Most Excellent Master, Royal Arch Mason), its primary color is red, and Royal Arch lodges are informally known as red lodges. This is all basic stuff, and for more information I recommend Mackey's Encyclopedia of Freemasonry (keeping in mind the basic biases that come from a document written by freemasons for freemasons, it still does an admirable job of covering the basics of what freemasons believe and the general divisions).

The Hermetic Order of the Golden Dawn was founded in 1888; the three principals - Samuel Liddell MacGregor Mathers, William Robert Woodman, and William Wynn Westcott - were all freemasons, and used the general form of freemasonry, including the hierarchy of members and the initiation rituals, the chartering of lodges, etc. It initially conferred five degrees. Unlike regular masonry, the Golden Dawn also allowed female members - a characteristic of some irregular brands called co-freemasonry, such as Le Droit Humain in France which was founded in the late 19th century.

Freemasonry was also a considerable influence on Theosophy, which is made especially clear by founder Helena Blavatsky in her book Isis Unveiled (1894). One of the characteristics of Freemasonry is the assertion of its ancient character, and the lodges hold their authority from their line of descent, initiate to initiate, from a quasi-mythical origin (occasionally supplemented by an old charter, which gives a veneer of legitimacy). The same basic concept applied to both the Hermetic Order of the Golden Dawn and Theosophy, except they claimed authority (and legitimacy) by asserting that they were backed by semi-mythical groups - in the case of the Golden Dawn, the Cipher Manuscripts which described the degrees et al. were received from a freemason scholar, and a charter was supposedly given by a Rosicrucian group through their contact Anna Sprengel; the Theosophists claimed their authority from the Great White Brotherhood, sometimes called the Great White Lodge - a group of "Ascended Masters" that acted as a "mother lodge" to all lesser occult groups. As with the Golden Dawn, the Theosophical Society (and related groups) often formed into lodges.

Dion Fortune was a member of both the Theosophical Society and Alpha et Omega - an offshoot of the Hermetic Order of the Golden Dawn - and had various other contacts, memberships, and initiations; she basically inherited all of these filtered and occult versions of lodges - and she created (or fictionalized) the concept of the black lodge in "The Return of the Ritual" (1922) in The Secrets of Doctor Taverner. It's important to note here that Fortune doesn't define the idea of a "black lodge" - letting context suggest that they are unauthorized, or populated by black magicians, but the idea had currency in occult circles - for example, Helena and Nicholas Roerich wrote in Hierarchy (1931):

Secondly, in the case of an announcement, the black lodge may incidentally be informed. The presence of even one outsider can serve as an intermediary. The entire world is divided into black and white ones. Some serve consciously, others according to their nature, and the third present a jelly-like mass unfit for anything. The black lodge is strong, because for the combat with Light a powerful potentiality is needed. It is unwise to underestimate the forces of the adversaries, especially when their beloved Kali Yuga comes to its end. Certainly it is a decisive battle and one should take care that temptation and seduction do not touch the weak ones. The location of the main seat of the lodge of the dark ones was indicated long ago.

So whether or not Fortune invented the concept or simply popularized it, the concept was easily grasped and promulgated. The black lodge has since become, if not a staple of occult fiction, than a recurrent element. So that's the background of the lodges of various colors, in brief.

So I was curious - how popular were these groups at the time, and what are their lasting legacies?

Freemasonry was very popular during the late 19th/early 20th century (and even earlier), though exact numbers are hard to come by; this is partially seen in the proliferation of different types of freemasonry and the inflation in number of degrees, especially in the late 19th century. Many members of various occult groups (such as the founders of the Hermetic Order of the Golden Dawn) were also freemasons, and the work of those groups - especially in the collation and publishing of various materials - strongly influenced a considerable amount of modern-day occult practice, and through that fiction - even H. P. Lovecraft makes reference to the Theosophists in "The Call of Cthulhu," even if he gets the details wrong - for more on that see Robert M. Price's article HPL and HPB: Lovecraft's Use of Theosophy. It is difficult to give the scope of influence of these groups, because they are so varied and because they were so influential, finding traction in everything from roleplaying games to popular music.

And how closely does the worldview in Twin Peaks echo their beliefs?

General inspiration and symbolism more than anything in-depth, from what I judge, but people literally speculate and analyze each episode in-depth. Certainly Lynch & co. don't depict the White Lodge as literally a bunch of ascended masters sitting around directing the occult currents of the world, or the black lodge as a bunch of self-destructive black magicians constantly thwarted in their designs, as Fortune had it.

Fantastic summation. Thank you immensely!