You probably knew all this already, but it never hurts to cover the basics. Now that I know that you know what to do when there is an emergency we can move on to the more interesting question of what actually constitutes an emergency.
Fortunately, true emergencies are much less common in pets than in humans. If you look at the eight most common emergencies in people - chest pain, stroke symptoms, accidents, choking, abdominal pain, seziures and shortness of breath - really only the last two are at all common and easy to recognize in pets. They do get abdominal pain, but it's harder to tell and is fortunately less often life threatening (no appendix in there to burst). Dogs and cats rarely have strokes and even more rarely have "heart attacks". In fact, coronary artery disease is unknown in our pets. Yes, they do get other kinds of heart diseases, but these tend to be chronic and do not often result in a sudden worsening constituting an emergency. True choking (i.e. not coughing or gagging that sounds like choking) is also less common than you might think. And pets do have accidents, but far less frequently than people, maybe because they don't drink or drive or ski or cycle or take showers or clean their guns or play with matches or rewire their homes or try to create viral videos...
As an aside, when I started in practice in the early 1990s "HBC" was a fairly regular emergency presentation. This had nothing to do with the Hudson's Bay Company, but rather it is our abbreviation for "Hit By Car". These days far more dogs are on leash and far more cats are kept indoors, so we may only have a handful of HBCs a year. Similarly, "BD-LD" is on the decline. Can't guess? "Big Dog - Little Dog", which is a traumatic dog fight injury where the size and strength differential leads to serious wounds in the "LD". We still see this, but people generally seem to be more aware of dog behaviour (generally - not universally), and again, more dogs are on leash. That being said, the increasing popularity of off-leash dog parks is preventing BD-LD from declining as quickly as HBC. Cat fights are far less common though than they once were. (Unfortunately we do not have an acronym for those.)
So now that you know what not to worry too much about, what should you worry about? When should you call "Pet 911"? The AVMA has provided a useful list. I will summarize an amended version here:
1. Severe bleeding or bleeding that doesn't stop within five minutes.
2. Choking, difficulty breathing or nonstop coughing and gagging.
3. Inability to urinate or obvious pain associated with urinating.
4. Eye injuries.
5. You suspect or know your pet has eaten something poisonous such as antifreeze, xylitol (in sugar free gum), chocolate, grapes, rodent poison, etc.
6. Seizures and/or staggering.
7. Fractured bones, severe lameness or inability to move leg(s).
8. Obvious signs of pain or extreme anxiety.
9. Heat stress or heatstroke.
10. Severe vomiting – more than two major bouts in a 24-hour period, or combined with obvious illness or any of the other problems listed here.
11. Refusal to drink for 24 hours or more.
I worked in an emergency clinic for a little while after I graduated, which is a story unto itself, and I can tell you that 90% of what called and came in was not on that list. But that's absolutely ok. A good emergency service provides peace of mind. They can often triage on the phone whether your pet needs to be seen or not. Consequently I can give you a greatly simplified list of when to call:
1. Your pet appears to be in distress (or, conversely, very lethargic).
2. You are in distress about something regarding your pet.
Don't hesitate to call. You're not bothering someone. It's their job to help and they are happy to do it. Unless you are drunk and it's 2:00 am and you want to ask why your cat is staring at the wall (true story). Then reconsider.