Clubs are a distinctive and important part of wealthy London life. A club provides its members with a place to rest: to eat, drink, sleep (like a private hotel), smoke, or simply chat with fellow-members. To become a member, one must apply and provide the names of members -- usually two -- who will support your candidacy. There is then a vote by the members, and unless a certain number (again, usually two) of members cast a black ball against you, you are admitted. You must then pay an annual fee to keep up your membership.
As a rule, new members pay an entrance fee, then a much smaller annual subscription rate. Often members who do not live in London pay a good deal less, the theory being that they make less use of the club.
Clubs generally have a character, a sort of person who can be found there, and clubs are chosen on that basis. Thus the AthenaeumClub? is for senior academics, and the CarltonClub? is for members of the ConservativeParty or its wealthy allies. There is often clear overlap, and unless there is some reason a club bars membership in certain other clubs one can be a member of as many as one likes -- providing always that one is willing to keep up all those subscriptions.
Membership in clubs is commonly restricted to gentlemen, though there are clubs that accept lady members and a few for ladies only.
Here is a list of clubs -- this is not all of them, by any means, but it's a good chunk: