Not much is known about their history. The Uber Ethereal revealed that they have conquered numerous other worlds across the universe for the purpose of genetic engineering the perfect psionic species.
The Aliens are comprised of at numerous species.
Ethereals (a.k.a The Elders)
The commanding species and outside of the Chosen, the only ones that appear to have free will. They are very tall and emaciated looking bipedal humanoids with four arms. Their strength comes from their high psionic power and ability to control minds.
Three generals and incredibly powerful warriors. Implied to have once been human siblings once, they were genetically modified until they resemble light-purple-skinned humanoids with glowing orange eyes. The Assassin wields a katana and shotgun-like beam cannon for close-range skirmish. The Hunter wields a sniper beam rifle for long range assault. The Warlock mainly uses his psionic mastery to mind control people and only uses his assault beam rifle when there are no other options.
The weakest species, these resemble traditional Grey aliens. In XCOM: Enemy Unknown, they're about four feet tall with large heads and eyes. They mainly act as forward scouts. Sectoid Commanders look similar, but have larger heads with prominent veins and are capable of mind controlling humans. In XCOM 2, human DNA was introduced into their genome, making them about five and a half feet tall with more human-like proportions.
Their infiltration specialists, they are disguised as lanky tall men with off-putting body proportions and movement. In XCOM 2, they no longer use their disguise, revealing them to be a reptilian species that resemble snakes with two weak arms that can carry weapons.
Aerial combat specialists. They consist of the armored top half of a bipedal alien and move through the use of jetpacks.
Bulky front-line troops. They're about seven feet tall with purple skin and heavy musculature. They wear green powered armor and wield plasma rifles and grenades. Muton Elites wear red armor and wield heavy plasma cannons.
Heavily armored melee troops that specialize in charging through heavy fire. Initially a Muton specialization, they're made into a separate species in XCOM 2. The new species is about twice as large as the Mutons with red and white musculature and lots of implants all over their bodies.
Insect-like aliens with six limbs that walk on four legs. They reproduce by injecting venom into live humans, who will then die, become a zombie-like creature for a few minutes, and then burst apart to reveal a new full-grown Chryssalid. In XCOM 2, they instead create cocoons out of freshly killed subjects that then produce multiple new Chryssalids. They are mainly used in terrorist attacks against civilians.
After conquering Earth, they replaced the world's governments with ADVENT. Their military consists of human collaborators that were genetically-modified into a new hybrid species. They mostly appear human except for their faces, which have smaller noses and significantly larger eyes. They hide these changes with large helmets that only show their jawlines. They wear heavy powered armor and wield either beam weapons or stun batons.
A shapeshifting species capable of assuming human form. In their natural form, they appear to be tall, heavy mounds of loose skin. They are used as spies.