It’s unlikely that you will hear a radio distress call but hams should know how to respond to one. Hopefully you will never need to send a distress call yourself but it’s best to know how to call for help in an emergency.
While some situations may be questionable, if life/health/property are genuinely threatened, it merits a distress call. Basically, if you would want police, fire, or ambulance response, it’s a distress scenario.
Obviously if one is in town with a telephone nearby, the proper response is to call local emergency services (in USA, dial 911). But if the phones are down or someone is very remote, radio may be the only way to communicate.
Distress calls are formally done by voice mode using the term Mayday-Mayday! SOS is the CW mode or Morse visual equivalent of Mayday; you are even less likely to encounter it.
If you hear a distress call:
First of all, stop whatever you are doing (QSO, or whatever) and cease all transmissions to avoid interfering with the distress call or a response.
Second (very important!), write down everything you hear related to the distress call. Under duress, people tend to forget things and stress will be high at both ends of the radio. Details about the incident are crucial for sending help. Include the date, time, frequency and mode.
Third, see if anybody else responds. Perhaps there is another station in a better position to react (location, facilities, etc.) If nobody else does, briefly ID and ask what is the nature of the incident. The distress station may not have a call sign or know proper radio procedure; be prepared to work with a non-ham at the other end. If time permits, confirm the details you wrote down with the distress caller.
Fourth, determine how to help the distress station. You need to get them help, but that depends on the location and nature of the trouble. If unclear or uncertain, contact local emergency services and let them help you sort it out, even if the distress is far away. In any case you must relay important detail such as the type of emergency, how many people are involved, and the distressed party’s location. The Mayday caller may also provide specific guidance if they are trained and knowledgeable.
Lastly, maintain contact with the distress station and emergency responders until help has arrived and the danger is passed.
Know also that in a genuine emergency/distress situation, you can violate normal protocol and rules to assist another station, at least in the USA (probably most other countries as well). If it requires using a mode, frequency, or power level for which you are not authorized, do it!
To send a distress call should you ever need to:
If the phones are down, use your VHF/UHF radio on a local repeater or national simplex frequency where other hams are likely to copy you (choose the most active local repeater).
If you are very remote and there is no chance any line-of-sight radios can pick you up, you’ll need a HF rig and antenna. Use the most suitable band for calling Mayday or SOS, considering time of day and propagation conditions.
Whether locally on VHF/UHF or more distant HF distress calls, call “Mayday-Mayday” with ID. Wait for a response, then repeat until someone replies. Consider your power source; if limited to small batteries, don’t call too often or for long.
Once someone replies, give essential info only. At minimum, nature of emergency, how many people are involved, and location.
During VHF/UHF or HF nets or ordinary contacts a ham may report an urgent situation by transmitting “Emergency” and their call sign
Another commonly-recognized emergency interruption is to transmit “break-break” plus your call. Upon hearing “emergency” or break-break” other hams should cease transmission and someone (NCS in case of net) should ask for info from the distress station.
By the way, the use of phonetics is essential for precise, reliable communication during a distress situation. Another good reason to know and practice the ITU phonetic alphabet.
In summary, we hope you will never have to send or respond to a distress call, but you should be familiar with them and know how to deal with either situation.
Some useful references about distress calls:
Ham’s Life simple, understandable
Distress Signal History interesting and detailed
Stack Exchange practical detail