The ways an animal can express illness are limited. The underlying causes for those expressions of illness are a tangled web that can often need a vet to unravel.
SYMPTOMS REQUIRING AN IMMEDIATE CALL TO THE VET:
1.) Horse refusing to eat, pawing the ground, kicking his bwelly, looking around at his flanks, not defecating, defecating only small amounts, rolling, getting up and down, sweating.
2.) Deep wounds exposing underlying tissue or any deep puncture to the abdominal area.
3.) Any wound that is gushing blood and especially any wound that is spurting blood like a fountain indicative of an arterial would (no matter how small).
4.) Three legged lameness.
5.) Depression (not interested in surroundings, not drinking, not eating) and lethargy with refusal to move.
6.) Depression with refusal to eat.
7.) Fever with nasal discharge, dropped head, labored breathing.
8.) Depression with labored breathing
9.) Ataxia (walking and standing as if drunk)
10.) Refusal to move
IF YOUR HORSE EXHIBITS NON LIFE THREATENING SYMPTOMS THAT LAST MORE THAN A DAY, CALL THE VET!! Some illnesses may seem mild but can become serious if let go. Some of these have a high cure rate early on, but can become chronic and disabling if not treated early.
Regular Veterinary care is very important to horse ownership. Things you need to do even if your horse is healthy are annual vet exams including stools samples. The amount of testing stools for worms varies with the type of worming program you are on and what your vet recommends. Back in the dinosaur days there were not the worming products that are available today (Ivermectin was a huge breakthrough) but that does not alleviate the necessity to worm regularly and to have stool samples checked.
Vaccination is also important, especialy if you are competitive. This protocol should be discussed with your vet. More experienced people vaccinate their own horses but in some States where Rabies is endemic, the State may require all rabies vaccines be given by a licensed vet (NY is such a State).
I don't mean for this to come off as "preachy" (tho it probably has.. you know old people 'get like that' LOL).
Please feel free to add to this list and if I am wrong about any of this, do not hesitate to add something saying so.. after all.. my knowledge is sometimes dated. I have tried to keep up over the years.. but I never have stopped learning!