Angel Emblems are magical talismans granted to Angels in Heaven. The talismans are in limited supply so must be earned through some sort of testing. The testing is unspecified but does not appear to be strictly combat related as non-Warrior Angels like Kinako can earn them as well. The Emblems are manufactured in Heaven by "artisans", like Darius Elexion. The crafting process for each Emblem is described as being long and difficult to master which is why the supply is limited.
All Emblems have a magical enchantment placed on them which prevents non-Angels from wearing, or even touching, the Emblems. The Holy Protection Enchantment is not necessary to the Emblems' function and can be removed so that even Demons can use them. Only an artisan Angel has the necessary knowledge and magical skill to perform such an alteration though.
The Sun Emblem is the most basic Emblem, and the first that all Angels must earn. Angels who have earned this Emblem are called, simply enough, "Sun Angels".
This Emblem gives the Angel their wings; however, it is mentally controlled, the Angel can conceal their wings if they wish. Kieri Suizahn has said that it also "calms the hearts" of Angels; this can be interpreted as the Emblem promoting emotional self-control to remain calm in the face of trouble.
Sakido received her Sun Pendant from her foster father, Darius. Her wings had been seriously injured when she was very young which prevented her from flying. Putting on the Sun Emblem caused them to spontaneously regenerate (rapid regrowth) and enable her to fly. The wings she received from the pendant seem to possibly be better than her natural wings would have been. When Buwaro puts this pendant on, he grew wings as well, but the effect is much less successful on him as they are too small to fly. It appears that Demons don't have mental control over the effect like Angels do, this makes sense since the wings are actual blood and bone.
The Star Emblem is the second Emblem that Angels earn. It grants an Angel their Halo around their head but only whilst it is mentally switched on.
This Emblem improves an Angel's Magic power. It is supposed to help improve "concentration" which can be read as determination, willpower, or even mental clarity.
Buwaro received his Star Pendant from his foster father, like Sakido. Buwaro's injury before he was born caused him to defensively enter into the Demonic Berserker rage as soon as he was born, this problem has never healed so he must wear the pendant to suppress it. Like the Sun Emblem, the Star Emblem's effect is constant, Buwaro does not seem to be able to switch it off even if he wanted to. He also doesn't gain the Halo effect. As a secondary effect, even if Buwaro did enter his Berserk State he would still have control of his actions. The first instance of a Berserk Demon maintaining sanity was actually with Sakido as she first touched then held the pendent which kept her from attacking Buwaro.
In one of Kieri's flashbacks it is revealed that she partook the test to obtain it but failed and the fact became another of her mother's verbal abuses towards her.
The Moon Emblem is the final Emblem in the sequence. Only Angels who already have their Sun and Star Emblems can earn this one. Angels, or Warrior Angels at least, who have all three Emblems are called "Seraph Angels".
This Emblem simply increases the Angels physical strength and endurance. It does not appear to have a visible effect like the other two and may actually be passive instead of switched on and off like the others.
Iratu received his Moon Emblem from his foster father as well. Unlike the others, Iratu didn't have an injury that needed to be healed but he was physically stunted for an Earth Demon. Healthy Earth Demons are normally 12 feet (3.6 meters) tall, Iratu was stunted because he was separated from his parents whilst still young (Demons nurse their young, they don't need to eat to survive but they do grow bigger, stronger and healthier if they do). When wearing the Moon Emblem, Iratu's size increases drastically, becoming the biggest Earth Demon in Hell. If he removes it, he shrinks to his normal size, though he will apparently experience growing pains when doing so.