That is really something you should ask The British Federation of Film Societies ... but let me suggest a few key elements. My practical knowledge is out-of-date and I trust some of our experienced readers will offers corrections and additions. - Dave Watterson.
These are important because they give film societies the right to exist - just. We slip in despite legislation aimed in the first place at dealing with the fire hazards early cinemas represented, then the moral hazards that sex cinema clubs offered to the pure British public. The secret is almost in a double-negative damned if I can remember where somewhere in the 70th clause or thereabouts! Essentially because we meet as a members' club, relatively infrequently we escape most provisions of the Acts. And local authorities cannot control us as they would sometimes like to do.
These are worth a close read. Basically they are licences to screen the film x times in place y and do not guarantee to lend you a copy of the movie! They are not renting the prints or DVDs out no matter how often everyone uses expressions like that. In fact they do lend you a copy of the movie to show.
If you rent a lawn mower you can earn money cutting all the lawns in the street. But if you "rent" a film, you buy the right to show it once usually in a specific hall and cannot show it as often as you like in as many places as you like.
The trade used to have many complaints from cinemas about unfair competition hence the tight controls, limits on advertising and so on.
My own view is that a film society depends as much on the social aspect as the cinema one. It often works well in a community where people have moved out of an urban environment for the tranquility of rural or small town life. Like-minded individuals with an interest in cinema can usually find enough others to band together and rent a few films for a test season.
The more clubby it feels the more people will muck in with the work and costs. Using a membership system encourages people not just to pay up for one show but to commit themselves to a half or a whole season of films.
Many local authorities have cinema provision in their local plans but no chance of attracting commercial companies sometimes film societies can get reduced rental of authority premises in such cases, perhaps even help with publicity. With a few notable (?noble) exceptions film societies work best where they offer a mix of films: mainly new but with occasional golden oldies, mainly English language but with a leavening of subtitled work. (In Bradford on Avon we can push that to a season which is half English-language and half other languages.)
Very few specialist societies survive. Those devoted to westerns, films noir and so on rarely last more than a season. The Fantastic Film Club and Vintage Film Club are about the only real exceptions I know. A few fan groups keep others going such as Laurel & Hardy fan groups (chapters of "Sons of the Desert" in their lingo.)
Though few societies try it there does seem to be a market for specialist screenings of old classic films - Metropolis, Birth of a Nation, The General (Keaton) and so on as a sideline. Larger societies sometimes subsidise such screenings.
Societies which meet in cinemas risk losing social cohesion and hence continuity.
Student film societies are a special and fascinating case. On the one hand they often have a captive audience but on the other it is an audience with many other attractions provided and the societies suffer because of the fast turnover of committee members. That said many of the best remain at the top of the heap for generations.
The absolute essential for a good film society? One enthusiast who can inspire others. S/he need not be a leader type but their presence is essential to stimulate others into doing their bit. The most worrying problem? It's not changing formats, the advent of video etc. those are just mechanical and cost problems it is the difficulty in finding people prepared to commit themselves to working for the society more than one season. It is easy to get hard workers for an evening or two but they will not promise to be there for every show.
For video screenings under society conditions DVDs offer the best source material but the success of a presentation depends on the video projector and sound system in use. Sound has long been the neglected element of film society screenings. There are honourable exceptions. Some of the university societies have built amazing sound systems capable of making the seats quiver. But all too often you hear a muffled mutter from a speaker placed on the floor near the screen. Aaaargh!