A random encounter is a common RPG situation. It most often occurs when the player character is traversing a distance across the world map or a dungeon, resulting in a conflict with a randomly chosen opponent. In Fallout and Fallout 2, random encounters are only encountered on the world map and can be as simple as a trader wishing to show you their wares, to finding a tin man who needs to be oiled...
Not every encounter is hostile. Not every dangerous encounter involves aggressively minded NPCs (such as "Stampede" or "Dehydration" encounters in Fallout). Not every encounter even has an NPC on its map ("Old battle" encounter). Some random encounters are unique, and happen only once during the course of the game (or not at all) - these are commonly known as special encounters.
The kind of encounter depends on many factors. The main one is player's location on the World map. Others may be the player character's Outdoorsman skill, Karma level or reputation, as well as their previous actions during the game.
Encounters on Desert squares depend on the player's location on the world map (split into 13 areas based on towns and other landmarks). Encounters on Coast, Mountain and City squares have less categories to choose from.
All terrain types primarily use the player's Luck to determine encounter type, as well as if the player is a Childkiller or passes Outdoorsman checks. (the skill checks only applies to certain encounters)
Several encounters do not appear in the un-modified game. These are usually "bad" encounters such as large groups of Super Mutants. It is not clear if this was intentional or an oversight.
The random encounters in Fallout 2 are determined by the player's location on the worldmap, his level and global variables governing it and finally, the chances of encountering one, stored in WORLDMAP.TXT. They can be anything: a number of molerats foraging, a press gang or a traveling merchant caravan.
A random element is the place where a given event will appear. Once a special encounter has been activated in a particular area no other random event will occur in this area. You can save outside the spawning area and keep reloading if you don't get the encounter you want.
In principle, there are two different types of special encounters in the game: unique encounters and repeatable ones. Unique encounters will only occur once per playthrough whereas repeatable ones can occur multiple times. Another difference is that repeatable events can happen in any of the random encounter locations but unique encounters do not; there are 2 sets of unique events, each of which occurs only in a certain set of locations.