Deciding whether something is an emergency is hard precisely because most of the time, it isn't one. Here's the short list of things that genuinely don't wait for a normal appointment.
Breathing that looks like real effort
Open-mouth breathing in a cat, laboured or exaggerated breathing in either species, or breathing that looks effortful rather than just fast, needs a vet immediately, not a wait-and-see.
Straining with nothing coming out
Repeated attempts to urinate with little or no result, especially in male cats, can be a blocked bladder, which is a genuine emergency, not a wait-until-morning situation.
Collapse, seizure, or sudden weakness
Any of these, even briefly and even if your pet seems to recover, warrants an immediate call. What looks like a one-off can be the first visible sign of something that's been building.
A swollen or hard abdomen
Sudden bloating, obvious discomfort when the belly is touched, or unproductive retching, particularly in dogs, can point to a genuine emergency and shouldn't be monitored overnight.
Sudden change in gum colour
Pale, blue-tinged, or bright red gums instead of the usual healthy pink are worth an immediate call, especially alongside any other symptom on this list.
Everything else is still worth writing down
The vast majority of changes aren't emergencies, but they're still worth a record, since a vet's first question at a normal appointment is almost always "when did this start, and has it changed?" Halo's daily log and vitals tracking exist for exactly that, so the answer is a real record, not a guess.