The sad fact is that the US airline industry is required to offer relatively few passenger rights when something goes wrong. The US DOT has increased compensation for bumped passengers and lost luggage, but in other countries, airlines have to step to the plate when passengers are stranded, delayed, or otherwise mistreated.
If an airline changes the flight schedule
If the new schedule doesn't work for you, you can request a refund, but this may take a long time to get. Airlines love to keep hold of your money, and there are no federal rules requiring that they refund you at all or within a specified time period. Pay by credit card, and contest the charges. Unfortunately, you may end up paying more for a replacement flight or for extra hotel nights, or you may end up not using hotel space or a tour that you've paid for.
If the airline loses or damages your luggage
US DOT regulations do require reimbursement of up to $3300 on domestic flights (not international, where different rules apply depending on airline and routing). But airlines are allowed to depreciate the value of your bags' contents and to ask for receipts so most people never get the full amount.
If the airline "bumps" you (involuntary denied boarding)
Federal law does require that the airline compensate you. Under the rules, passengers who are involuntarily bumped will receive compensation equal to their one-way fare up to $400 if they are rescheduled to reach their destination within two hours of their original arrival time for domestic flights and four hours for international flights. The mandatory minimum compensation doubles to $800 if passengers reach their destination later than the two- or four-hour limits. Some loopholes apply, such as if the plane has fewer than 30 seats. Airlines usually offer flight coupons, but you're entitled to cash payment.
If your flight is delayed or you miss a connection due to a problem within the airline's control.
There are no Federal regulations stipulating compensation. Each airline deals with these situations on a case by case basis. In Europe, there are comprehensive regulations, which apply to US citizens (or those of any nationality) departing from the European Union (maximum compensation is 600 Euros for a long distance flight that arrives 6 or more hours behind schedule, unless the airline can prove that the delay was beyond its control; compensation varies depending on length of flight and length of delay). Canada has also established some basic passenger rights.
Most airlines will give you a refund, even on a non-refundable ticket, in cases where you will be so late to your destination that your trip would be futile or no longer necessary. This is not a Federal rule, but is spelled out in most airlines' contracts of carriage.
Some airlines will put you on another airline's flight in the case of a delay or cancellation, but there's no longer any rule requiring this (there used to be, under airline regulation). You just have to ask nicely and hope for the best.
If you're stuck on the tarmac for hours
No Federal law. Each airline handles this situation differently.
What passengers can do
In many cases, going to small claims court is the best route to get satisfaction. If the airline maintains an office or business in your small claims jurisdiction. Judges are airline passengers too, and many have suffered at the hands of the airlines, so are often sympathetic.
You can also join the Association for Airline Passenger Rights and lobby your elected officials for greater protection from what is largely an unregulated industry.
Get travel insurance and read the policy carefully. Don't buy directly from the airline, but from a reputable company such as Travel Guard or Access America.
If the value of your checked baggage is over the $3300 limit, buy excess valuation from the airline at check in. On domestic flights, you can buy up to $1700 in additional coverage for as little as $17 ($1 per $100); international airlines offer coverage for as little as 50 cents per $100, but the limits are lower, however international flights are not covered by the US DOT's $3300 upper limit (some airlines pay only $20 per pound in compensation for lost luggage, so excess valuation is a must for checked luggage on international flights).