“At that time, the year 2227, the Institute had made great strides in synth production. But it was never enough. Scientific curiosity, and the goal of perfection, drove them ever onward. What they wanted was... the perfect machine. So they followed the best example thus far - the human being. Walking, talking, fully articulate... Capable of anything.”— Father
Synths, created by the Institute, are cybernetic organisms designed to resemble humans. They are classified into one of three generations depending on their complexity:
Generation 1 synths are purely robotic androids, possessing all the physical capabilities of a human, but being no smarter than a standard protectron. They resemble a lanky skeleton with metal joints, mechanical "organs" and hydraulic muscles. They are completely loyal to the Institute and speak in a high-pitched, robotic tone. Like all robots, they can take large amounts of damage, never backing down from a fight even with their limbs blown off.
Generation 2 synths are similar in appearance to their predecessors, but are hardier and sheathed in plastic "flesh" mounted on a thin metal lattice surrounding their mechanical skeleton. Though more human-like in appearance, Gen-2s still possess limited A.I. and superficially resemble a mannequin more than a human, having visible seams along their sickly, yellow-gray skin. Their skin is fragile however, and will fall off in large chunks if struck or shot with enough force to reveal their mechanical insides. These synths are often found dressed in synth armor when on the surface, or else in Institute jumpers when working down in the Institute. They also speak in a robotic voice, but at a much lower pitch than Gen 1 synths.
Generation 3 synths represent the pinnacle of synth technology, being virtually indistinguishable from natural-born humans down to the cellular level. Each Gen 3 synth is built from lab-grown bones, muscles and other tissues that are assembled and brought to life at the Institute's Robotics lab, and are "born" with the bodies and mental faculties of full-grown adult humans. Though entirely biological, each Gen 3 synth contains a neurological implant inside their brain allowing them to be "programmed" and manipulated via voice commands. This implant cannot be detected nor removed without killing the synth.
Nick Valentine and DiMA are unique synths, both being prototype synths intended to bridge the gap between the Gen 2 synths and the truly sapient Gen 3 synths. While DiMA was made to freely develop a personality from experience, Nick's personality was built upon the mind of a pre-War detective who volunteered to have his memories encoded, giving him all of the memories and experiences of his namesake. The two of them escaped the Institute together, but due to the limited memory space of their prototype brains, Nick's memory of the escape was deleted. This led him to believe that he was thrown out by the Institute. DiMA later traveled to the Island and founded the synth refuge of Acadia with Faraday. As DiMA reached the limit of remaining space in his prototype brain, he began to artificially increase his memory capacity by storing his memories in connected computers.
Although it is unknown when each successive generation was developed, the people of Diamond City first encountered a synth in 2229, in an event called the Broken Mask incident. The synth, which called itself Mr. Carter, was a prototype that was field tested without the Institute director's approval. Though the Institute did not actually intend to massacre the people of Diamond City, the people of Diamond City have feared and resented the Institute ever since.
By 2277, at least one Generation 3 synth is known to have had traveled to the Capital Wasteland after becoming fully self-aware and escaping the Institute. With the help of the underground organization known as the Railroad, he had his appearance and memories altered and eventually wound up as the chief of security at Rivet City. Incidents of synths escaping have occurred multiple times prior to 2277 as the Synth Retention Bureau was created and had multiple coursers tracking rogue synths as far as the Island.
Many denizens of the Commonwealth have become paranoid and prejudiced against synths, fearing they might be abducted by the Institute and replaced with identical copies of themselves. Their paranoia is justified, as some of the most sophisticated synths encountered act as spies and could fool the unaware. There are some examples of Commonwealth citizens that were murdered and replaced with synth duplicates whom the Sole Survivor interacts with on missions if they choose to join the Institute.
The synths' Institute creators view them as nothing more than machines and treat them as slaves. The Railroad has committed itself to freeing Generation 3 synths from their masters and smuggling them out of the Commonwealth to somewhere the Institute cannot reach them. In contrast, the Brotherhood of Steel has sworn to eliminate all synths, for to them, a machine capable of thinking like a human is an abomination of technology.
Most encountered hostile synths belong to either Generations 1 or 2, are armed with Institute laser weapons and shock batons and are immune to radiation damage. When fighting against the Gen 1 and 2 synths, it is possible to shoot off their limbs. The moment one of their arms is destroyed, they may be unable to use their weapons. However, they will then proceed to rush the player character and begin attacking them with their remaining arm. Shooting their legs will result in them crawling to the player character and attacking them with their hands.
Gen 1 and 2 synths are less intelligent than Gen 3 synths, but display a level of sapience, as exemplified in their dialogue. They call out to their target "whoever you are, I know you're there" and will comment that their systems may be in need of recalibration if they lose track of a target. During the Tradecraft quest, Deacon comments that Gen 1 and 2 synths are on a similar mental level to protectrons and other pre-War robots, causing some debate among members of the Railroad as to if Gen 1 and 2 synths are worth trying to rescue.
Generation 3 synths are physically and mentally indistinguishable from ordinary humans, having lab-grown bodies of real human flesh, bones, and organs instead of plastic and metal. As Dr. Roslyn Chambers found, no medical tests or procedures can identify someone as a synth without killing them (synths that infiltrate settlements, for example, only drop synth components when killed). Psychological tests, such as Covenant's SAFE test, are dubious at best. However, they are not perfect duplicates, and have a few key differences from humans that are not immediately obvious: synths cannot gain or lose weight (one Institute researcher remarked on this after discovering that many of the Institute synths had a taste for snack cakes).
There are many Generation 3 synths that are hidden in plain sight in the Commonwealth. They include, but are not limited to, Magnolia, the singer of the Third Rail and Danse, a paladin of the Brotherhood of Steel.
While most Synths are Gen 1 or Gen 2 and are thus allied to the Institute, synths can also be encountered in other places.
If the player character has built a radio beacon, incoming settlers have the potential to be Gen 3 synths (non-infiltration). If these settlers are put on a supply line and happen to pass any major Brotherhood of Steel encampment (Cambridge, etc.) they will be fired upon by the entire station, stalling any building the player character can do. 
Synths can also be a part of a Minutemen patrol if the Sole Survivor has retaken the Castle and has an ample amount of well-off settlements. If a Minutemen patrol that possesses these synths passes a Brotherhood encampment, they will not be fired upon. Sometimes their patrol routes take them straight through the Boston Airport, into the courtyard of Fort Strong, and back through again. They will not fire upon each other and will sometimes pleasantly engage in dialogue with each other. 
If the player character has established a bad reputation with the Institute (in other words, completed the quest Banished from the Institute), the Sole Survivor's settlements will often be infiltrated by synths who will eventually open fire on the player character's other settlers.
This will be a rare occurrence after the Institute is destroyed.
Stronger than an ordinary synth, these striders, while fairly weak, are still a significant step up. They, like their counterparts, almost exclusively use Institute weapons and in some instances, shock batons.
A step up from the synth strider and leader variants, the patroller is stronger and has better aim. They are usually seen in groups with other synths, such as synth striders or leaders, and are usually seen with Institute pistols or rifles.
Synths appear in Fallout 3 and Fallout 4. The two synths encountered in Fallout 3, Armitage and Harkness, were ostensibly Generation 3 synths, and Generation 1 and 2 synths were not introduced until Fallout 4.
The following is based on unverified behind the scenes information and has not been confirmed by canon sources.
Synths share many similarities with the terminators from the Terminator film franchise. One being that the Generation 1 synths are nothing more than endoskeletons that loosely resemble a human skeleton, just like the majority of terminators created by SkyNet. The Generation 2 synths differ in that they possess artificial skin like the T-600 terminators shown in Terminator Salvation. The Generation 3 synths were designed to look exactly like human beings so that they can infiltrate human settlements undetected, which is the exact purpose of the T-800 infiltrator seen in every Terminator film. Sometimes when in combat with Gen 1 and Gen 2 synths, they will say to the player character "You must be terminated." Finally, the biggest similarity is that both the Fallout and Terminator franchises depict a post apocalyptic America destroyed by nuclear war.
The Generation 3 synths are quite similar to the Replicants from Blade Runner, based on the novel Do Androids Dream of Electric Sheep?, in that they are synthetic humans genetically engineered entirely out of organic substances and are virtually indistinguishable from real human beings. The author, Philip K. Dick explored ideas of different types of conscious beings as equal with humanity, and the human reaction to these rival sentient beings. This is one of many Blade Runner references throughout the game. In fact, the surname "Deckard," if used in some way as part of the player character's name, is one of the names that Codsworth is capable of saying. This is a reference to Rick Deckard from the film. In addition, the test required to enter Covenant measures psychological response to random questions, very similar to the Voight-Kampff test that Blade Runners give to suspected Replicants.
The hunting of rogue synths somewhat references the movie Artificial Intelligence. The movie depicts a futuristic America where androids are used as servants for various jobs, but they commonly gain self awareness and run away only to be hunted down and destroyed.
The exact distinction between Gen 1 and Gen 2 synths is a little vague. In the games files, there are three types of non Generation 3 synths, not two. Some are unambiguously Gen 1 synths and spawn with a completely exposed mechanical skeleton and typically do not use armor, as seen in Kellogg's memories during Dangerous Minds. On the other hand, some are unambiguously Gen 2 synths and have intact "skin" which can only be damaged by crippling their limbs, expressive faces, and typically wear clothes, such as the synth requisition officer. In between these are synths with rigid plastic skin, which typically spawn at least partially damaged and can be blasted off by the player character even without crippling limbs, similar to other robots. The last group could represent Gen 2's that have begun to decay, but given the more rigid appearance of their skin, they appear to be an earlier model. It could be that Gen 1 synths may have been built with plastic casings that often degraded over time and some units simply have none left.
Gen 2 synths bear similarities to the NS-5 robots from the film I, Robot.
Synths may be based on Star Trek's Androids. Both have human variants such as Data and Gen-3 Synths. There is controversy on what to call them, androids or synths, synths being short for synthetic humans. Both have controversy as to whether they are people or not, as Data, at one point, was threatened to be dissected and used to create more, saying he is not a person. Synths have the Railroad and East Coast Brotherhood of Steel, and they both have someone they call Father, for Data it is the scientist who created him and Lore. For the synths, it is Shaun, who was essential in creating 3rd Gens.
End of information based on unverified behind the scenes information.