Mayday in and of itself launches nothing except the pulse rate of the ATC controller hearing it. It’s what you ask for next that may trigger wider response.
If you announce your intention for an expedited return to the airport, or an emergency descent, you’re telling him/her what you’ve probably already started doing and are asking him/her to get anybody else out of the way. And if you need help finding your way someplace they can help with that too.
Once you’ve communicated the severity of your situation and what your intentions are, they may (ref Richard above) notify the fire department (Airport Rescue and Fire Fighting or “ARFF” in US parlance), call the FBI bomb squad, NORAD to scramble fighters, contact USAF or the Coast Guard to launch a search and rescue force, or none of the above. Their response depends totally on the specifics of the situation.
From my POV, if I’m going to need anything above or beyond normal handling to a typical airport or if I have any material safety concerns beyond ops normal, that’s an emergency and deserves a Mayday. One amber idiot light is not even a Pan. Until it degrades to something that affects control, navigation, or stopping ability.
As to administrative “emergencies” for legalistic reasons, the most common one is an overweight landing. There are a couple others.
Big airplanes often take off heavier than their normal day-to-day legal landing weight. If a prompt return to the takeoff airport is needed you face a decision: do you land now overweight, fly around as needed to burn fuel to reduce to max landing weight, or dump fuel overboard if you have that capability? For a long haul airplane, [fly around as needed to burn the excess weight] may be 5 hour job. Perhaps surprisingly not all long haul airplanes have fuel dumping. It’s also expensive and environmentally unfriendly.
The approved solution is to look up the landing distance required at the actual weight and if the runway is long enough, land now at your current weight. The airplane can safely land at the very max takeoff weight. The lower limit in day-to-day use is about long term durability and wear and tear. A truly crappy landing below max landing weight is very unlikely to do real harm. The airplane can take 20 years of typical crappy landings without bending. A truly crappy landing in an overweight situation might break something. And the routine volume of routinely crappy landings might bend something over time. The engineers set the weight limits based on that kind of lifetime durability tradeoff.
The syllogism runs like this:
Landing above max landing weight is a violation of the normal day-to-day regulations. In an emergency a pilot may violate regulations as deemed necessary to maximize safety. In the present circumstance, landing overweight is as safe, or safer, than noodling around for 5 hours waiting for something else to go wrong. So therefore to land overweight for a safety reason we must be in an emergency situation. So we must inform FAA of the emergency. So we say “mayday mayday mayday”. Yes, a bit silly. And definitely legalistic.
We (not me) had one recently that left Chicago going to Europe. Near Montreal they discovered all the toilets were stopped up. Something was done improperly in servicing. Oops. They can’t take 250 people another 8 hours with no outhouses; the sanitation crisis was already well underway when the cockpit found out.
Solution: They divert to JFK, declare a mayday because they’re landing overweight, make a nice touchdown and taxi to a gate. There the lavs are properly serviced. Meanwhile maintenance conducts the fairly minor overweight landing inspection and the aircraft is refueled. They depart again 90 minutes after they parked.
Absent the clogged potties there was nothing wrong with the airplane or crew at all. It’s an emergency because the lawyers say it’s an emergency.