I was doing some research on the Black Lantern crisis as part of my philosophy degree, and there were mentions of a bunch of Lantern Corps I'd never heard of? I knew about Green, obviously, and there's been members of the Justice League that've used Pink and White rings.
The ones I was caught on were specifically the Blue, Orange, and "Indigo Tribe" corps. I found a photo from during the crisis with the second Green Lantern and a handful of Justice League members in monocolour outfits, but the only source I could find didn't say anything about what those Corps actually do.
So, what are the Blue Lanterns, Orange Lanterns, and Indigo Lanterns? And how do they interact with Earth?
All of the Lantern Corps draw on something know as the Emotional Spectrum, an underlying energy generated by the act of living beings existing and feeling the things they feel. While this power is ambient and available to all the most famous users of this power source is the intergalactic Green Lantern Corps who tap into the green light of willpower. With this willpower they are able to fuel their rings, generating constructs from their own imaginations that can take any shape they can hold.
Created when the first living being moved under its own power, the green light of will is one of the oldest of the emotional spectrum only outdone in age by the white light of life itself. But as the millennia wore on and life became more complex this light continued to fracture and branch into each new emotion that life summoned within itself. The three you have cited are, in turn:
(A cool wallpaper someone made showing off all of the lantern logos that we're aware of)
Note that all of the information I am sharing here is cobbled together from what has been shared with the public by relevant sources, most often Earth's own Green Lanterns but also the wider Justice League and others.
The Blue Lantern Corps is the most straight forward of the three. Animated by the blue light of hope they are what seems to be basically monks on one hand and a planetary version of doctors without borders on the other. They appear to planets in distress, due to some kind of civil strife or natural disaster and use their abilities and knowledge to return calm. Known most readily by their mantra "All Will Be Well" they have even shown the ability to return an aging, dying star to its original vibrant state. They work most closely with the Green Lantern Corps, almost as a mercy corps to the GLC's sometimes harsh peacekeeping methods. While an individual Blue Lantern is not nearly as strong as their Green counterparts, when in close proximity BOTH receive a huge jolt in power. It also holds the useful honor of being able to drain the power from both Yellow and Red Lantern Rings, as hope eats away at both fear and rage, as well as being immune to being drained in turn by the Orange Ring of Avarice.
The Indigo Tribe, rather than being a mercy corps, seems to be more closely comparable to a penal battalion. Wielding the indigo light of compassion, it is the most elusive of the lights and the hardest to hold. Those that it claims for itself being made up of the violent, the hateful and the heartless. Opening itself to them they are overcome with empathy and remorse, offered a place among the indigo tribe to begin to make up for the pain they have caused in life. Giving up all personal belongings, titles, even their names and languages they become part of the Indigo Tribe itself. Travelling from place to place in a battle against the worst of the worst. They are seen as very enigmatic, and more than a bit creepy. They talk in a language that cannot be translated and their members seem to hold no thoughts or emotions of their own. Though they are entrusted with the potent ability to mirror and understand the emotions of any other living being around then, extended to an ability to copy, absorb and unleash any other Lantern energy turned against them.
The Orange Lantern "Corps", often pithily referred to by our own earthbound Lanterns as "Agent Orange" is animated by the orange light of avarice. As one would expect, SHARING said power isn't exactly conducive to this ring's abilities so more often than not there is only one Orange Lantern Ring active at any one time. It is held by a very...charming looking individual named Larfleeze, you can look up a picture of him on the Green Lantern embassy website if you like. He is motivated by greed and only greed. To own as much and as many and as widely as he can. This is reflected in the abilities of the ring, which allows him to create a copy of anyTHING or anyONE that his ring has consumed. So his "Orange Lantern Corps" is actually made up of constructs representing the other beings he has devoured in his swath across the starways. It also has the ability to drain the power from other Lantern Rings unless they are the Blue Rings of hope or the Violet Rings of love both of which are antithetical to the manifestation of greed.
These three groups are poorly understood here on Earth because they seem to be the rarest of the Corps. While the Green Lanterns have uncountable thousands or more members, dedicated to the dispensing of law and justice across the inhabited universe, and the other Lantern Corps of the Yellow, Violet or Red varieties have been attached to supervillains who have made at least a semi-permanent residency on Earth. These three groups (well, two groups and one weirdo) are much smaller pieces of the tapestry. The Indigo Tribe are enigmatic, only appearing where and when they need to. Larfleeze is out in space, presumably trying to steal something. And we already HAVE champions to keep our hearts filled with hope and our societies stable against the forces of darkness. It's just that our symbol of hope happens to be a big red S rather than a geometric design in blue.