It was created by the old Civil Aeronautics Board (CAB) before the days of airline deregulation. The rule was a paragraph in an airlines' contract of carriage - the legal agreement between the passenger and the airline - that states its responsibilities to passengers when a flight is delayed or canceled. Rule 240 mandated that an airline facing a delayed or canceled flight had to transfer the passenger to another carrier if 1) the second carrier could get you to your destination more quickly than the original line and 2) it had available seats. Before airlines were deregulated in 1978, all the big U.S. airlines adhered to this rule.
Does it even exist?
This answer is an ongoing feud in the travel industry. Some travel "experts" and even some airlines say it's a myth, while others say it exists. In today's deregulated environment, airlines don't have to post tariffs so technically Rule 240 no longer exists; hence, some people think Rule 240 is a myth. However, a majority of airlines still honor the old rules, including "240", and continue to use it, to the passenger's advantage, everyday. Plus, some airlines still have Rule 240 in their contract. For example, if you check on Delta Air Lines' domestic contract of carriage, you'll find something called Rule 240 that promises the airline "will exercise reasonable efforts to carry you and your baggage according to Delta's published schedules and the schedule reflected on your ticket."
How does it work?
This rule may be invoked if your flight is delayed or canceled for other reasons besides weather so that you can take another flight and still make it your destination. For example, if you're flying on United Airlines from Chicago to New York and you find out your flight has been delayed two hours, you can kindly ask the gate agent invoke Rule 240 for you. It's very possible that they will print you a boarding pass for a different carrier (space permitted) flying to the same destination.
How does it apply to me?
This applies to you because you have rights as a passenger that you probably never even knew about - everything from when you're entitled to a refund to what the carrier owes you when you're bumped from a flight. Some smaller carriers don't even publish their contracts online, meaning you have to ask for a copy of the document at the ticket counter. Under the federal law, the airline must show it to you. You should keep Rule 240 in mind next time your flight gets delayed or canceled by anything other than weather. Rule 240 still gets implemented even though it's not official.
What's the bottom line?
The bottom line is that while no one airline is legally mandated to follow Rule 240, many of them still do. The key is that you have to ask, not demand, and in many cases, you'll be accommodated.