The premise of the X-Files is that Mulder and Scully are FBI agents working on unexplained cases- the so called "X" Files. Many of these cases deal with extraterrestrial phoenomena (UFO sightings, Exrtaterrestrial Biological Entities...), and often involve government conspiricies that hide the truth about aliens. Mulder's background is in psychology; his psychological profiles of killers had him on the fast track at the FBI until he chose the basement office to work on the X Files- motivated by his personal connection with the search for aliens, the abduction of his 8-year-old sister Samantha which has always haunted him. Scully, on the other hand, is a medical doctor; her field is forensics, and she is often called upon to perform autopsies (at 11:21pm). She began her work with the X-Files in the pilot episode when she was assigned to scientifically explain away Mulder's theories and ideas- that is, to debunk the X-Files. However, she immediately grew to respect and trust Mulder ("you're the only one I trust" -EBE, Season 1) and immediately took on the role of full-fledged partner. She is portrayed as the rational one; she is grounded in reality and fact, while Mulder is regularly off testing a wild new theory.
There are two different types of X-Files episode: the mythology or 'mytharc' episodes, which deal with the overall plot, aliens, and the government conspiracy to hide the truth, and the "monster of the week" episodes. (Note that the latter may address killers, weird occurences, machines, etc.- very few actually deal with monsters.) Many are humorous (see "Hollywood A.D."- season 7, written by David Duchovny), and some treat the ever-developing relationship between Mulder and Scully, which occaisionally verges on the non-platonic. Since the first episode, there has been unresolved sexual tension (UST) between the partners, and the fan community (the "X-Philes") is divided on whether or not they should get together. Those in favor of the relationship are called "shippers"; those opposed are "noromos." The undecided fans reside on the "fence." There are hundreds of web pages devoted to the discussion/argument of this relationship; the most common argument against it is that a Mulder/Scully Romance (MSR) would ruin the tension that has been some of the greatest intrigue of the show. Many fans of this persuasion identify as "finishippers"- those who believe Mulder and Scully should get together, but only at the very end. (I speculate that CC is a finishipper...) Personally, I'm a devoted shipper and this page will be biased thus.
Some of the most important developments in the ongoing plot of the X-Files (not to mention the MSR) occured in Fight the Future, the feature film made in 1998 based on the television series, starring (of course) David and Gillian. If you'd like ALL the details, you may download the script. I recommend highly, however, that you rent it- if nothing else, the hallway scene is so frequently referenced in Phile pages that you will be lost without it. It will become immediately obvious to which scene the references "hallway" and "bee" are made.