1. Difficulty Urinating, Cannot Pass UrineThis would seem obvious that this would be an emergency situation from the pain aspect alone, but sometimes people assume that the animal has voided elsewhere or somehow has bladder control when in fact it may be a physical blockage. This is a condition most commonly associated with cats, but can occur in dogs as well. Male animals are at the most risk, since they have a long narrow urethra. The most common cause of obstruction is a stone or silt. Other causes include cancer or inflammation.
Signs to watch for: straining to urinate, licking genital area, bloody urine, painful abdomen, painful when picked up, not eating, vomiting or diarrhea. Time is of the essence; urinary obstruction/bladder rupture can be fatal in a matter of hours.
2. Eye That is not Normal or is InjuredEye conditions are almost always considered urgent, with some notable exceptions such as cherry eye or tear staining. With eyes, things can happen very quickly, and a quick treatment can be sight-saving in some cases.
Signs to watch for: squinting, blinking more than normal (blepharospasm), rubbing or pawing at the eyes, redness, discharge, etc. please don't delay in calling your vet to be seen asap. Time can make all of the difference in some cases.
3. Vomiting or DiarrheaVomiting is fairly common, especially for cats (hairballs). A change in diet can produce soft stools that usually resolve quickly. Attitude is of key importance -- an animal that is vomiting or has diarrhea and is weak, depressed, and does not have an appetite is a more serious and possibly emergency situation.
Signs to watch for: Seeing blood in vomit or diarrhea is also indication of a potential emergency. Some diseases such as Hemorrhagic GastroEnteritis (HGE) appear suddenly, without warning and features bloody vomiting or diarrhea are true emergency situations.
4. Foul DischargeMalodorous discharge from a wound or body opening is a sign of infection, and is something that most often needs to be seen right away. One possible exception would be puppy vaginitis, a common finding in young female dogs that is usually self-limiting.
Signs to watch for: excessive licking, red, inflamed skin, old wounds, foul odor, poor appetite, lethargy, increased thirst or urination
5. Physical TraumaPhysical trauma can be anything from being attacked by another animal to being hit by a vehicle. This is another one that would seem obvious as an emergency, but many times the animal will run away (or to their owner), without noticeable wounds. Hopefully your pet is fine, but the effects of adrenaline can be powerful, and initially they may seem more fine than they actually are. It is always important to make sure that there isn't a concussion, deep bite wound, internal bleeding, or broken bones. Signs may take a few hours to 48 hours appear.
Signs to watch for: pain, difficulty moving, pale gums, difficulty urinating (or unable to produce urine), coughing.