NOTE: MASSIVE SPOILERS FOR PROMETHEUS, ALIEN AND ALIENS.
An Engineer ship crash-landing on the surface of LV-223 in 2093.
Time and Date
Prometheus opens in 2089 with the discovery of a cave painting in Scotland which points the way to the Engineer base. The ship arrives at this location in the final week of 2093, with the final moments of the film taking place on New Year's Day, 2094.
No date is given in Alien for the action, save that it happens in the 22nd Century (due to the presence of a crew uniform patch that says, 'Flag of the United Americas 2104 to present'). In Aliens Carter Burke orders the colonists to investigate the crashed Engineer ship on 12 June '79. Assuming Aliens happens in 2179, then Alien takes place 57 years earlier, in 2122 (and this was later confirmed in featurettes in the Alien Legacy boxed set). From a computer display at the start of Alien, the movie starts on 3 June.
Thus, Prometheus concludes 28 years, 5 months and 2 days before the start of Alien.
The planet LV-426 orbits, along with several of its moons, in 2122.
The planetary body that Prometheus flies to is called LV-223. The planetoid that the Nostromo crew land on in Alien (and is later colonised by the Weyland-Yutani Corporation prior to the events of Aliens) is called LV-426, informally known as 'Acheron'. The different designations seemingly confirm that these are different planetoids.
Both planetoids are depicted as moons circling larger gas giants. LV-426 is one of at least four moons orbiting a red-hued gas giant. LV-223 is one of two moons orbiting a blue-coloured gas giant. Given that we physically see four moons in Alien (three moons and the gas giant are seen in LV-426's sky) and a comprehensive 3D starmap in Prometheus only shows two, the conclusion is that these are different gas giants (otherwise the gas giant changes colour and acquires two additional moons in thirty years, which seems implausible).
In Alien, LV-426 is identified as being located in the Zeta II Reticuli star system. Zeta Reticuli is a real star system located 39.16 light-years from Earth in the constellation Reticulum, consisting of two stars in a binary orbit. However, the two stars are extremely far apart (dozens of times the distance between the Sun and Pluto), meaning that each star could hold an extensive solar system of its own without gravitationally interfering with the other.
LV-223, along with its mother planet and another moon, as shown on the Prometheus's scanners in 2093.
In Prometheus, the destination star system is not identified. A distance of 327,000,000,000,000 km is given, which translates as 34.56 light-years. Given that Zeta Reticuli's distance has been estimated with a strong degree of accuracy (the error margin is only 0.1 light-years), this would seem to confirm that LV-422 is not only a different planetoid to the one in Alien and Aliens, but is located in a totally different star system altogether. Some fans have postulated that LV-422 is located at Gliese 86, a star just under 35 light-years away in the constellation of Eridanus. This is especially popular as an extrasolar gas giant was discovered circling Gliese 86 in 2000. Gliese 86 and Zeta Reticuli are located in the same general neighbourhood, only being separated from one another by 10 light-years.
This splendid theorising has been torpedoed by Ridley Scott saying straight-out that Prometheus takes place in the Zeta Reticuli II star system as well, however.
Thus, the two planetoids in Prometheus and Alien - LV-223 and LV-426 - are different planetoids but they are located in the same star system. Based on the evidence above, I'd still suggest they are orbiting different gas giants.
The USS Sulaco approaching LV-426 in 2179.
Engineer bases and ships
According to Prometheus, the Engineers built an extensive military installation on LV-223 more than two thousand years ago. This installation consists of approximately five large domed buildings, each huge in size. At least two of the buildings had large, horseshoe-shaped spacecraft located adjacent to them. The installation appeared to be a base for the creation of a biological weapon of mass destruction, apparently for use against Earth. This facility was overrun and its population almost completely wiped out by unknown forces (but likely a bioweapon they lost control of) approximately 2,000 years before the events of Alien.
In Alien and Aliens, an Engineer starship of similar design to those seen in Prometheus is found on the surface of LV-426. Initial assumptions were that it had crashed, but more recent interviews (at the 18-minute mark) have suggested it landed or was parked deliberately there. According to Ridley Scott, this ship originated at the LV-223 facility and was on its way somewhere else (presumably not Earth) with its cargo of facehuggers when its cargo got out of control. The pilot landed on LV-426 and was killed, within a couple of hundred years of the destruction of the LV-223 facility (so between 1,800 and 2,200 years before the events of Alien). The fact that the facehugger eggs could survive and remain viable for that time period is impressive.
A mural in the Engineer base on LV-223, suggesting that the xenomorphs were extant more than 2,000 years ago.
The bioweapons and the xenomorph
On LV-223 a black liquid stored in vase-like containers serves as a destructive bioweapon. It can animate corpses, turning them into monstrous killers, and transform little worms into large, snake-like monsters. Rather more bizarrely, it can convert human sperm into a parasite-like creature that, when given a female human body to gestate in, transforms into a squid-like creature which can grow to colossal (some might indeed say, totally fricking preposterous) size and then impregnate another type of creature into another host, a creature which more closely resembles the traditional xenomorphs.
On LV-426, the cargo of the crashed Engineer ship consisted of eggs which, when hatched, produced parasitic 'facehuggers'. These creatures would attach themselves to a human or animal host and place an embryo in their chest. After a period of gestation (typically several hours, or several days for a queen creature capable of laying further eggs en masse) this 'chestburster' erupts through the host's ribcage and grows to large size within a matter of hours. This creature is the traditional xenomorph. Unlike the black goo things on LV-223, the xenomorph's life cycle appears fairly stable and predictable.
Note that, based on both the information provided by Scott in interviews and the mural in the LV-223 facility depicting the traditional xenomorph, the traditional xeno appears to have already been in existence for some time when the base on LV-223 was wiped out. This would then seem to contradict the popular (and perhaps obvious) theory that the black goo stuff in Prometheus is some type of prototype that would lead to the familiar xeno in future films (though the appearance of a proto-xeno in the final seconds of Prometheus would seem to suggest that this was the direction things were heading in).
Based on all of this I would argue that the standard xenomorph was already in existence and the Prometheus bioweapon was an attempt to replicate it. Given the inefficency of the Prometheus creatures, with a confusing and bizarre life-cycle, it can be concluded that the Prometheus bioweapon was a miserable failure. Perhaps all of their 'normal' xenomorph eggs had been put on the LV-426 ship and they were forced to develop a secondary weapon when their main one was put beyond their reach (which seems extremely unlikely, but there doesn't seem to be too many other conclusions that can be reached)?
The Engineers are an intelligent alien race who may have had a hand in the appearance of life on Earth. If not, they certainly visited our stone age ancestors around 35,000 years ago. 2,000 years ago a group of Engineers, possibly military in origin, established a base on LV-223, a moon in the Zeta II Reticuli system, 39 light-years from Earth. They created a bioweapon, apparently taking inspiration from an already-existing alien lifeform known as the xenomorph. They apparently decided to wipe out life on Earth for reasons unknown (possibly ranging from fear that their creations were getting out of control to one of their emissaries being nailed to a cross - this latter idea is extremely idiotic, so hopefully that's not the direction they are going in).
A ship took of from the LV-223 base carrying a cargo hold full of xenomorph eggs. The pilot ended up getting infected. He made an emergency landing on LV-426, a moon circling a neighbouring gas giant in the same system, but was killed. He activated a warning beacon telling his fellows to stay away. They respected that and did not go after him. Instead, they decided to use their own bioweapon (perhaps thinking they could control it better than the xenos themselves, or perhaps they had put all of their xeno eggs on the ship and lost them in the crash) against Earth, but it got out of control and wiped out most of the facility. The last surviving Engineer managed to seal himself in stasis in a ship away from the threat of the bioweapon but ended up oversleeping by 2,000 years, until he was awoken by the crew of the Prometheus and was then infected by the bioweapon and killed.
There are still plot holes you can drive a power loader through in this scenario, but this does seem to be a fairly likely chain of events given the information we have so far.