Di Gi Charat is too broad of a topic to really summarize easily, so I'm going to cheat here and send you over to its Wikipedia page.
As for the Black Gema Gema Gang, I've taken valuable hours out of my life in order to write an informational (is that a word? *laugh*) document on the foursome. Buckle up.
-Summary-
The Black Gema Gema Gang (Burakku Gema Gema Dan in Japanese) is a group consisting of their leader, Pyocola Analogue III, the vet, Rik Heisenberg, the dentist, Ky Schweitzer, and the doctor, Coo Erhard. There are also many creatures called “Gebas” and Nazo Gema.
Though considered the “enemies” in Di Gi Charat, the group as a whole is actually rather harmless, and the members have good hearts. They’re also rather gullible, often having their own traps backfire into their faces.
The gang stars in a spinoff series called Leave It To Piyoko! AKA Piyoko ni Omakase Pyo! (can also be translated as "Let Piyoko Help Pyo"/"Piyoko can Help Pyo"). This series is a two-part manga as well as an eight episode anime.
Though they have the same title, they differ very much in the plots and storylines. The manga tends to be more “cute”, which the anime seems to aim at “funny” (though this is not to say there aren’t any funny moments in the manga or cute moments in the anime). Except for episode 6 of the anime, most of the stories have nothing to do with the stories in the manga.The Gang also leaves their planet for different reasons. In the manga, Piyoko is heading straight for earth on a mission to capture Dejiko and the boys of the gang show up later. In the anime, the whole group flees the planet after Piyoko accidentally blows up a city with her special attack. They run away to avoid trouble and to get money in order to restore the city in addition to restoring the planet.
They’re from “Planet Analogue” (Anaroogu Sei), which happens to be very poor. In the Leave It To Piyoko anime, a flashback from 5 years ago depicts it as being a considerably wealthier planet. They’re currently poor due to a war with their twin planet, “Planet Di Gi Charat” (De Ji Kyaratto Sei) that took place some time in the past. Analogue’s leaders, who are simply referred to as the King and Queen (Piyoko’s parents), tried to take over Di Gi Charat by force (a move that Coo claims was dumb anyway). The war failed and Analogue was left poor.
Their goal throughout the various series of Di Gi Charat is to capture the main character, Dejiko. According to the Leave It To Piyoko anime, this was original Ky’s idea. Because Dejiko is the princess of planet Di Gi Charat, they reason that she could be held for a large ransom. They never do succeed in this mission, although they do come close on a few occasions.
The gang first appears in the four-panel comic series “From Gamers” by Koge-Donbo. The first anime episode they show up in is the Di Gi Charat Christmas Special.
-PKO-
Within the Black Gema Gema Gang, there is a subgroup known as the “PKO”, or “Pyocola Keeping Operations”. This is Rik, Ky, and Coo. There’s really no significance to this subgroup, except that it excludes Piyoko, the Gebas, and Nazo Gema. They are meant to protect Piyoko.
All three boys wear collars and have dog tails. It is uncertain whether or not their tails are real, as there are a few pictures depicting them with out tails, but the tails are also often seen wagging. Rik and Ky have dog ears, and though Coo is usually seen wearing his dog cap, he is sometimes depicted with similar ears as well.
It’s note worthy that the days of the PKO boys’ birthdays are in a row. Ky’s is March 3rd, Rik’s is April 4th, and Coo’s is May 5th.
Also, this could be a coincidence, but ironically, in the Hoshi no Kirby games series, Kirby has three animal companions (in addition to others): Rick the hamster, Coo the owl, and Kine the fish. Their names, spelled in Japanese, would be Riku, Kuu, and Kain, and aside from Kine’s “n”, these are the exact same spellings of the PKO boys’ names.
Yet more pointless name trivia; in the anime, "Ground Defense Force Mao-chan", the three main girls each have power-ups entitled "Rikubo", "Kuubo", and "Kaibo" (rikubo deals with land defense, kuubo with air, and kaibo with sea, if memory serves correctly). Going back to the Kirby thing, Rick helps on land, Coo in the air, and Kine in the water, so most likely, these names have to do with elements (in which case, Rik would relate to ground, Coo to air, and Ky to water...)
Also, all three PKO boys have German last names. There was a German scientist named Werner Heisenberg, a German politician named Ludwig Erhard, and a German philosopher, humanitarian, and physician named Albert Schweitzer. Whether this is significant or not is unknown.
-Families and relationships-
Like in most anime, the families of the Black Gema Gema Gang are almost never discussed or mentioned. Piyoko’s parents are the only ones that are ever actually shown (in flashbacks of Rik and Ky, who explain what they look like to her), but they are only shown in the anime. They look absolutely nothing like her, sort of resembling sock puppets. Whether or not this is truly how they appear remains a mystery. It has been speculated by fans that it’s possible that her parents died when she was little, therefore she does not properly what they look like, but this is most likely fanon.
The only other relative of the Black Gema Gema Gang ever mentioned is Ky’s mother, who supposedly gave him the cup he wears. Why she gave it to him, when she gave it to him, and if she’s still alive is unknown.
Rik’s parents are probably no longer alive. He never mentions them. Same applies to Coo’s parents.
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The Black Gema Gema Gang is very popular for “shippering” among the fans. Here are some of the relationships among the gang and some information on them. (I'm very biased, keep in mind)
-Piyoko and Coo-
Arguably the most popular Di Gi Charat pairing (aside from Usada and Minagawa), this is also probably the most canon. Koge-Donbo points out in a commentary that it is one of the only real pairings in Di Gi Charat. These two have a great deal of interaction, specifically in most manga works written by Hina (who drew Leave It To Piyoko, Piyo Piyo Piyoko-chan, and a few various stories in the official Di Gi Charat anthologies). This pairing is also largely hinted at in the sixth episode of the Leave It To Piyoko anime (which is based on a story from the manga in the first place, although they differ in many ways). It’s alluded to that Coo has a crush on Piyoko several times, but she seems to be too innocent to notice at her current age.
-Piyoko and Rik-
Taking into consideration the extremely large age gap between these two, this pairing is entirely fanon. Their relationship is platonic, and neither show any outward affection towards the other that isn’t done in a father-daughter-like manner.
-Piyoko and Dejiko-
Entirely fanon. Their relationship is platonic. Piyoko does claim to “like oneechan a lot” but this is only because Piyoko likes her as a big sister.
-Piyoko and Ky-
Most likely the only Ky-related pairing that could be considered to have any “evidence”, although he does obviously obsess over Piyoko, this is a fanon pairing. Ky loves her like a daughter as does Rik.
-Ky and Dejiko-
Fanon, since these two have minimum interraction, but there is a minority of fans for this pairing, and one of the artists in the official Di Gi Charat anthology even noted the two as being her favorite pairing.
-Ky and Rik-
Spurred, most likely, from Piyoko referring to Rik as her dad and Ky as her mom. Very popular among the female fans of Di Gi Charat, but this pairing is fanon. It may even be implied in Di Gi Charat Nyo that the PKO boys are homophobes. A male character called them “beautiful” (although he did not mean it the way they took it), and the boys panicked, grabbing Piyoko (who probably didn’t get it) and running away.
-Rik and Dejiko-
Though mostly considered fanon, these two do get some interraction in the Leave It To Piyoko manga, when Dejiko asks Rik to help her save an injured kitty. He helps the cat and Dejiko thanks him, and he replies, “I did it for the cat, not you.” However, considering the large age gap (and Rik’s undying loyalty to Analogue), this pairing is very unlikely.
-Rik and Akari-
Akari Usada, a character from Di Gi Charat Nyo, looks a bit like Rabi~en~Rose, though is older. It is believed that the two girls are cousins. Rik and Akari don’t ever have direct interraction, but because Akari is a famous popstar, Rik is aware of her, and even cosplays as her briefly in an episode. He also offers an Akari make-up kit to Piyoko to cheer her up, and it is often wondered by fans of this pairing why he had the kit on hand in the first place. Fanon.
-Rik and Usada-
Fanon. These two don’t have any interraction.
-Coo and Ky-
See Ky and Rik.
-Coo and Usada-
Fanon. The only interraction that has been made between these two was in an episode of Di Gi Charat Nyo. The gang is trying to trick Usada in order to upset Dejiko, so Rik gets the idea of putting a fake love note in her locker. Usada’s parents spot Coo outside of their house and jump to the conclusion that he sent the note. But because it is already canon that Usada has a relationship with Minagawa, this pairing is extremely unlikely.
-Coo and Dejiko-
Fanon. The only interraction between them is: 1. When Piyoko “traded” the PKO in return for Dejiko acting like her big sister for the day, and Dejiko orders him to “come closer to me”, and: 2. Dejiko teasing Coo for being shy about treasuring the panda he received from Piyoko.
-Coo and Kyuara-
Kyuara, a bear-girl (not 100% certain if that's what she is) and fellow classmate of Piyoko and Coo in their Evil Prep School (see Piyo Piyo Piyoko-chan in Appearances) has a crush on the “Major” of the Black Gema Gema Gang, but doesn’t realize Coo and Major are the same person. While she doesn’t take any direct interest in Coo, the fact that she is a fan of the Major may lead one to conclude these two could become a pairing, along with the fact that Coo is deliberately nicer to her than Piyoko in this series. However, Kyuara is a fangirl of Major despite never having met him (most likely, this is a crush based on looks, for she also points out that Coo looks like the Major), and Coo usually acts nice towards Kyuara in order to get a reaction from Piyoko. Though this pairing’s not fanon, it’s not canon either.
-Nazo Gema and Gema-
It’s uncertain whether or not this can be classified as fanon or canon, but several artists in the anthologies (Koge-Donbo included) stated that they would like to write love stories involving these two, and it is believed that there is a story about them in the second volume of the Leave It To Piyoko manga.
-Seiyuu-
The well-known seiyuu Megumi Hayashibara voices Piyoko. She has done many voice acting roles as well as songs, and is a certified nurse. Many anime fans consider her the “Queen of Seiyuu”. Her ability to capture Piyoko’s cuteness is remarkable.
Tomo Saeki voices Coo. She’s done many roles in various anime, mostly in shoujo. She has a good vocal range. She voices Salt from “A Little Snow Fairy Sugar”, who bears a strong resemblance to Coo (although this is partly because both were designed by Koge-Donbo.)
Kousuke Toriumi voices Rik. He’s done minor roles in various anime and video games. His deep, serious voice fits Rik very well.
Chihiro Suzuki voices Ky. He usually does side characters, but he’s had a few major roles. Humorously, he voices Megane-kun (AKA Dexter) from Tottoko Hamutaro (Hamtaro). Ky and Megane are both bespeckled and share uptight personalities.
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