The first thing is to make sure your horse is up to date with his vaccines and dewormers. Depending on the country you live in, some vaccine are obligatory or just recommended.
Then, do a "visual check-up". Does you horse's hair look bright? Do his feet seem healthy? Does he eat well and drink enough? Everything that doesn't seem normal should alert you.
You horse should also see the dentist once a year to check his teeth growth.
I personally show my horse to an osteopath once a year too.
If anything seems abnormal, you can take your horse temperature with a thermometer. A horse temperature is around 37,5-38 °C at rest.
Feel free to call your veterinarian if you have any doubt about your horse's health, he is the best person to anwer you.