This Emerald is a Gem that ended up left on Earth when the Gem War came to an end, and as such remained there and eventually discovered by the Crystal Gems. He started off cracked, but his Gemstone got healed by the start of Season 3.
Emerald was made a lot smaller than other Emeralds, about the size of Peridot, and his hair is designed in a spiked punk style, with his outfit mainly mirroring Earth punk bikers.
Upon initial discovery, Emerald's stone was found to be cracked. He was yellow-green in skin color, with his hair a standard green. He had an eyepatch over his eye, with a black leather coat, dark-grey shorts and black grey boots. Since his stone was cracked, he was mute, and also had a permanent smile on his face. This form is briefly shown again during Season Eight.
After his Gemstone was healed, his appearance changed slightly: his hair became more lime green compared to the dull green, and his skin turned into a similar color. His clothes turned from black and grey into dark green and yellow green to match his Gemstone. Some other extras were added to his outfit: spikes on his boots, and a pair of shades over his eyes colored similarly to Peridot's visors.
Emerald is a cheeky type of Gem, and one that enjoys messing around with others when he can. This sort of behavior is mostly used on Peridot, enjoying her reactions to his antics and finding them funny. Later on in the series this behavior is extended towards other members of the Harmony Gems. His time on Earth made him appreciate what Earth has to offer, and has grown accustom to humanity and how things work, sooner than even the Crystal Gems themselves. As such though, he doesn't usually involve in the Crystal Gems' missions too deeply, often going out and enjoying life on Earth even if he is tasked directly to said missions by the others.
His time as a cracked Gem also made him more sympathetic towards other Gems going through similar defects. This is shown a number of times, even towards Harvested Gems such as Sessile, or more dangerous Gems such as Rupee.
It is numerously hinted on that he used to be far more cold and ruthless like most Emeralds are during his time in the Gem War, having a far more aggressive behavior towards others. This partially is due to the pressures of being a defective Gem, as he himself states that he had to put in more work to prove his worth as a Captain.
As revealed throughout Season Eight, Emerald was once apart of the Gem War as one of the Homeworld Captains under Hessonite's command, pointed out by other Emeralds as being very full of himself, but got left behind when The Diamond Authority released the Corruption Light onto Earth, with a possible reason being his "defective" state. It is unknown how he managed to avoid becoming corrupted, nor how his gemstone became cracked.
He appeared during the first days of winter, being found by Peridot after he threw some snowballs at her. He was later found by the Crystal Gems, but he was later released. He stocked up on more snowballs afterwards, expecting her to follow him, but instead he was pinned down by Kyra, questioning why he was being brutal to one of the elites. Kyra told him he was lucky punishment was already given to him due to the gemstone being cracked.
When Kyra had Discord make a portal to Homeworld, emerald saw his chance and managed to get himself through back as well. On his arrival, he managed to find the diamonds during the duel between Rupee and Jasper, and tried to warn them of Kyra's presence. He just managed to send one of his disks to use as a mirror to expose Kyra. Emerald wasn't overly involved during the Osicone attack.
Upon returning to Earth as a Crystal Gem, he got his gemstone healed at the Rose Fountain. He began to mess around with Peridot, and states that he would've stayed on Homeworld, yet they had a quick fix to handle his cracked gem, which was why he stuck around.
During his stay he began to make a living for himself away from the group, something Peridot caught attention of after a game of "lava on the floor" with him, Steven Universe, Ruby, and Sapphire. He flew off to Bayburgh and went on through his daily routine in his stay: checking out Bull's Eye Books n' Vids, and going to the market. During the market, he came across Sapphire, who he called "Sophie" for the sake of her disguise. He decided to invite them to a upcoming party that was gonna be held that night, to which Sapphire accepted, and he had them meet him at the Seaside Hotel. He revealed upon arriving that he knew the whole time that they were his friends, saying it was hard not to recognize a "square red block, a green pizza piece, and a blue cotton candy". Still, he went on to bring them to the party and didn't help them back until the next day.
When they found Discord's Body Puzzler, he ended up taking the body of Jasper, something which he deeply enjoyed throughout his time. This did make him become a bit cocky however towards Jasper (who had Peridot's body), which came to bite him later once he got his body back.
When Amethyst and Steven were going by Bayburgh, he came up to them and got them to go to the Brittle Bear Bakery with him. He claimed it was for a Gem monster, though it turned out to be a urban exploration video he wanted to do, as it was a trend on TubeTube he wanted to take a crack at, as many had done videos on the place anyway.
When he and the others were brought together during the Pink Diamond case, he began to question the lack of involvement most of them actually had in the war, coming to the shocking conclusion that they barely had any involvement at all with the exception of Jasper. In their attempts to make Pearl talk, he came up with the idea to drug her into talking, which ended up back-firing on them. He and the other Gems were captured by White Diamond and rescued by the Humans.
White Diamond Arc
His biggest involvement during this arc is in the chapter "Emerald's Quest", which he was assigned by Jasper to go to the White Tail Woods to handle a local Timberwolf population for Twilight Sparkle. During this investigation, he stumbled onto a trash site with all of the Harvested Gems stolen from Homeworld by Grogar's Army. He decided to go down and check out the place, discovering Sessile there as part of those stolen by Grogar. After learning more about her, and after his encounter with Trotter, he successfully takes Sessile away from there, wanting to bring her back to the barn rather than to Homeworld. On his way back, he briefly got involved with Jade and Flint on their way to deal with the feud between the HoofFields and McColts.
During the Era 3 Festival, Emerald got into a challenge with Aquamarine to prove who was better in the shooting gallery. This competition quickly turned into a fight between the two, Emerald unwittingly calling Aquamarine "girlfriend", which made Star Quartz believe he actually was. He and Aquamarine later find out that they were set up for an arranged marriage, which the two spent the rest of the day together trying to get out of it, only to find it was a lesson to teach them how much their behavior effects other people.
After Grogar's complete takeover of Homeworld, he was one of the gems placed under Jasper's training to better prepare everyone for Grogar's Army. During this, he went with Peridot, Flint, and Pearl back to Homeworld in order to scout out the area, though mainly it was to get a break from Jasper's classes. He was captured by Nichrome and taken back with the others to "Tambalon", getting his gemstone cracked again while he was there during the marking process adopted from Aredox. He did manage to escape and heal himself sometime after this.
My Little Universe II
After many of the Harmony Gems got rejuvenated thanks to Spinel, he personally became Jasper's superior as Emeralds were made to take charge over Quartzes. Mainly this was also the reason he decided to go after Jasper alongside Fulgurite when Hessonite drove some of their friends away, mainly as Jasper will only listen to him for that very reason.
In the chapter "Battering Ram", he and Fulgurite find Jasper and her destructive training within the Northern Forests, both Gems trying to convince Jasper to return with them but to no avail. When Fulgurite suggested them to just make her come back, he quickly refuses saying that she's not nice like him. During their attempts to do this, he and Fulgurite were found by Rupee, and was driven away as Rupee wanted to fight Jasper. Rather than stop the fight, he actually recorded the fight for his "TubeTube channel", much to Fulgurite's annoyance. After hours of the fight though, he intervened and suggested to the two fighters to make a contest to decide who was the better Gem. He and Fulgurite gave them numerous challenges to do, many involving fights with different monsters throughout Equestria. During the numerous fights, he later finds out about Rupee's severely damaged gemstone and begins to have some sympathy and worry over her, given his own experience. When they tried fighting the Ursa Major, Emerald quickly tried getting them all to leave, him specifically trying to make Rupee go away from them all the while before Luna, Starswirl the Bearded, and Fluttershy arrived on the scene. He was mainly the one that helped Rupee out once the fight was over.
She joins the Harmony Gems in both the final court session with Discord and White Diamond and the final fight with Grogar's Army. Before the fight, she remains behind in the Realm of Corners to buy her friends time to destroy Grogar's ships. However, in the end she was captured along with the other Harmony Gems and Off-Colors, and brought onto the ships to kill them in front of Empire City. Lucky for her she was freed by the recovered Harmony Gems and with their help they stop Grogar's forces from leaving Homeworld.
Emerald is able to move objects with his mind, though he mainly uses this on his Gem weaponry rather than on other things. With this Telekinesis he can actually fly around on his Gem weapon platform with little effort.