Summary

• Search the Web for your airline's special meal options.
• Preferably request a special meal immediately after booking your flight.
• Bring some backup food if you have dietary restrictions. Sometimes, special meals never arrive.
• If there's an upgrade, itinerary change, or missed connection, contact the airline.
• If you have dietary restrictions, confirm your special meal request three times. Confirm at check-in, confirm at the gate, and confirm on the plane before takeoff.
• Look at the labels and read the ingredient lists. A "vegan" meal might contain whey, from cow's milk. A "gluten-intolerant" meal might contain wheat. Other special meals might have other problems.
• If a "kosher" meal is not properly sealed, it may not actually be kosher anymore.
• If you get a box of raisins instead of a special meal: Politely ask staff if they could please try to find some more food which you can safely eat. After the flight, you can politely request whatever compensation you hope for: e.g. future flight credit, or a $20 airport meal voucher. The airline has no legal obligation to give you anything.

Details

Bring some backup food

The caterer or airline may fail to fulfill your special meal request. You might just get a banana and an apology from the crew. (Source.) If you have dietary restrictions, it's wise to bring some extra food onto the plane with you, just in case.

When booking your flight

1. Check if your flight includes free meal service. If your flight does not offer this service, you'll need to bring your own food.
2. If your itinerary includes a codeshare flight, or flights with multiple airlines, please post a new question.
3. Do a Web search to find out which special meals your airline offers. Some airlines may offer few or no special meal types.
4. It might be useful to find out the four-letter code for the meal type. That way, for extra certainty, you can confirm your meal request using both the name and the meal code.
   • Kosher is KSML. Lacto-ovo vegetarian is VLML. Vegan is VGML. A gluten-intolerant meal is GFML. There are other codes, too. Not every airline offers every special meal type.
5. Request your special meal. The earlier you do this, the better. (Source.) Ideally, make your request immediately after booking your flight. Otherwise, try to request it at least 72 hours before the flight. (Source.) You can do a Web search to try to find instructions on how to do this for your airline. If you can't find any advice, contact the airline directly.
   • If you request a special meal less than 72 hours before your flight, your request may or may not be fulfilled.

On the day of the flight

1. If possible, bring some backup food. Special meals sometimes don't make it onto the plane.
   • A large bag of roasted sunflower seed kernels would be a healthy, filling, energy-dense choice. After landing, please declare that you've brought food into the country. (Source.)
2. If there's a change of carrier, a flight schedule change, or an upgrade, contact your airline. (Source.)
3. If you miss a flight or connection, contact your airline, but your special meal will probably be gone. (Source.)
4. After checking in, look at your boarding pass. If it shows your special meal code, great! If not, quickly talk with an airline staff member, to make sure that your request is still in the system. (Source.)
5. Before takeoff, politely tell the gate agent and tell a flight attendant that you've requested a special meal in advance. (Source.)
6. If all went well, your special meal will be served to you at mealtime. Enjoy!
7. If your special meal is incorrect or unavailable: Politely ask a flight attendant if they could please look for some food which you can eat. If the found items are too small, politely ask for more.
   • After the flight, you can politely request whatever compensation you hope for: for example, an airline credit voucher, or a $20 airport meal voucher. Keep in mind that the airline has no legal obligation to give you anything. (Source.)

Warnings

• Please look at the labels, and read the ingredient lists on your special meal.
• A "gluten-intolerant" airline meal may contain gluten. (Source.) Maybe you can eat the fruit, plus certain other parts of the meal. (Source.)
• A "vegan" airline meal may include a small container of yogurt. (Source.)
• A "diabetic" airline meal may be loaded with sugar. (Source.)
• If a "kosher" meal is not properly sealed, it may not actually be kosher.
• Other special meals may have similar problems.
• So, again, please bring some backup food.
• If you order a special meal, you may become ineligible for free upgrades. (Source.)
• Even though these meals are called "special", they often don't taste any better than regular airline food.

Please don't inconvenience the staff by requesting a special meal just for fun. On the other hand, if you need a special meal for a good reason, please do request it. (Source.) [Disputed; please discuss. Please see also this comment.]

Edit

If I missed anything, please edit this answer, or leave a comment below.