Pseudo-random numbers are numbers that are being generated with some algorithm that makes it possible to repeat the sequence if you seed the generation algorithm with the same value as you used initially, but that still seem to be random in that it is incredibly hard to predict the sequence without knowing the algorithm, and that have statistical properties that closely mirror random numbers.
Elite is almost entirely based around pseudo random numbers: The universes are generated by seeding a random number generator with a fixed value. The planet names, political and economical data are generated by seeding the random number generator with a seed extracted from the random number generator with the universe seed.
Change the universe seed, and the game changes, completely.
Thats what allowed the original Elite to run on machines like the C64, the BBC, and a ton of others - contrary to many modern games, they didn't have to store ANY data about the planets, only a few bytes for the universe seed and a small set of functions to generate the code.
More modern examples of pseudo random numbers in computer games is Civilization and all it's descendants, Sim City and it's descendants and many other map based games.
However randomness could be used so much further. Games like civilization only create a basis for the further game, then AIs take over. Elite conceptually went much further, by basing the entire structure of the universe, including it's civilizations, on randomness, allthough it manifested itself in the game rather crudely (it didn't really affect you that the planet you visited was inhabited by green felines)
The main appeal of pseudo random numbers, or for that matter any formula or algorithm that can reasonably quickly generate unpredictable but repeatable sequences is that it allows you to represent a much larger world than you could ever hope to design or store on the users system. It also allow you represent a world with much larger amount of detail than would otherwise be practical.
It's worth noting that pseudo random numbers is usually only used to seed specialized generation algorithms for generating various part of the world.
For a world that is so large and complex, it makes sense that you don't need to model all events via AI's either. Most complex games, including Civilization for instance, hide a lot of what is going on from you. In most Civilization style games, you can only see what's going on around your own units, and even then there might be further restrictions on what you see.
In Elite, you only see whats going on immediately around your ship.
These restrictions play right into your hand if you want to use a randomly generated world.
Many of the benefits of randomness go straight out the window if you need to keep track of changes to account for developments in the world. However, in many types of games you don't need to most of the time.
Take an Elite style game, for instance, and add detail: Government types changes, military control and dominance fluctuates, etc. For almost any kind of change, you can put together an equation that give fluctuations over time that are reasonable. Add a few elements to the equation where you can affect it by random variables, and you have a model for generating a world where you can generate not only every position in space, but also in time.
Depending on the level of detail you want, you can go in at an arbitrary time and an arbitrary point in space, and see roughly what went on (in a newspaper headline type of way - details of events, such as how a ship navigates, will likely seem weird to a human player).
Now you have a static world that can be regenerated from a small string of numbers. But what about the player and non player AIs? Instead of recording everything that happens, you can calculate most of it, and whenever a player does something, you store information about deviations from the randomly generated world.
Imagine that you as a player in a space strategy game, destroy a ship. The ship was part of a military fleet trying to conquer a world. As a result of your action, the fleet loses its advantage, and the game doesn't manage to compensate by weakening the opposition quickly enough (without making it too obvious for the player), so the invasion fails, and the timeline doesn't work out anymore.
What should the game do? Record a deviation, and determine a target time for returning to the timeline. The world is under government X instead of Y, but some time over time T, the game will create events that will bring things back to where it should be.
As long as it's done smoothly enough, the players won't notice. This model also has the advantage that if the player stays close, then the player keeps affecting the timeline and will see little of the effects of the game slowly compensating to bring things back in sync, and if the players move around a lot, the player won't see whats going on, and any logical flaws in the compensation that is being done will be hidden.
This idea is essentially based on one of many theories trying to resolve the problems of time travel: What if any changes you make to the past will be compensated for so that you can only cause disruption for a limited period of time, before at least most of the rough differences between the two timelines have been resolved. So the further back in time you go, the less effect you can have on the present.
Structuring a world via pseudo random numbers in both time and space indeed make the game itself time travel in a way: The entire timeline of the game was set before the game started, and you as the player travel back in time and are allowed to affect limited pockets of space time, but as soon as you turn your back, the game engine compensates away all your changes, making you just a ripple in the ocean...
Wish I had time to do game programming