Brett being transformed into an Egg.

There have been two alternative reproduction cycles that have appeared in the movies that show how individual xenomorphs can reproduce without the need for a Queen: -The directors cut of Alien: There is a famous scene (deleted from the theatrical release) which shows Ripley discovering a barely-alive Captain Dallas and the body of Brett. They have been cocooned by the xenomorph, and are both mutating into one of the original eggs capable of creating a facehugger. This shows that a single xenomorph can use a victim for food and/or to continue the egg -> facehugger -> chestburster -> xenomorph -> egg life cycle. -Aliens vs Predator: Requiem: The Predalien (a xenomorph born from a Predator) is shown in this movie to have the unique ability to use it's second set of jaws to implant embryos down a victim's throat, thus bypassing the egg and facehugger phases and allowing a Predalien to directly spawn other xenomorphs. It's not specified where this ability comes from (possibly from the DNA coming from the Predator host). Xenomorphs are incomplete DNA wise, they evolve and advance by assimilating the DNA of the host. So while individual aliens take on the characteristics of the species they were incubated in, the race as a whole follows the normal life cycle of egg > facehugger > full alien, with "queen eggs" making "queen facehuggers" etc. This is one of the reasons Predators loved using Aliens as prey. 1) They changed with each new species they are exposed to and 2) they sit dormant in their eggs for as long as necessarily. In the novel Prey (Alien vs. Predator Book 1) we are occasionally taken into the minds of the predators. The main predator thinks to himself that when a bunch of xenomorph males are in close proximity to each other, one of them will morph into a queen. The first Alien vs. Predator movie is loosely based on this novel. 
Aliens vs. Predator - Prey - cover
This is presumably what also happened in the movie Aliens. The colonists investigated the ship and a number of xenomorphs were spawned. The xenomorphs would takesome of the colonists back to the derelict spacecraft and more xenomorphs were spawned. Eventually the all male population produced a female, which took up residence in the warm and humid atmospheric reactor. 
ALIEN queen layin eggs
  Cocooning as seen in the game Aliens Vs. Predator 3 is that the Xenomorph Number Six, after entering cocooning its victims, can change its appearance, eventually becoming a queen itself. This rapid genetic D1981 (talk) (Contribs) 02:17, September 5, 2014 (UTC)regeneration is also another evolutionary trait of the Xenomorph to ensure its very survival as a species. In Alien vs. Predators: Extinction the praetorian acted as a princess of the hive who then became the queen. This stage is seen with Number Six. If this is so, then Hudson is correct with them being like bees. Earth examples:A sex-determination system is a biological system that determines the development of sexual characteristics in an organism. Most sexual organisms have two sexes. In many species, sex determination is genetic.In other cases, sex is determined by environmental variables (such as temperature) or social variables (e.g. the size of an organism relative to other members of its population).Some species do not have a fixed sex, and instead change sex based on certain cues.