How are horses put down?

is not uncommon. Although often perceived as cold or impersonal shooting is a very rapid method of euthanasia. It involves no suffering and no further stress to the horse than lethal injection.  
Theres no right or wrong method it depends on individual preference and the situation at the time. If a decision needs to be made as an emergency there may be little choice. But its advisable to have an idea of what you might choose should the need arise. Some owners find euthanasia by shooting more distressing. Particularly owing to the noise and the fact that they are required to be more distant from their horse. 
What happens to my horses body afterwards?
If the horse has been euthanised by lethal injection you can arrange for incineration with or without recovery of his ashes as a keepsake or you could choose to bury him. Bear in mind that burial can involve significant costs time and require special approval. 
Horses euthanised by the free bullet method could later be incinerated buried or potentially allowed to enter the human food chain. Only horses fit for human consumption may enter the food chain. This means the horses passport must indicate as such in section 9.
However in many cases the horse will have been signed out of the human food chain automatically because of medication that it has received during its lifetime preventing this from being an option. Additionally to be fit for human consumption the horse would need to be sent to a licensed slaughterhouse for euthanasia and management of the carcass something which may deter many owners. 
Either way you can make these arrangements yourself or alternatively your veterinarian can make them for you. Its probably best not to be around when your horses body is removed because it can be distressing to see. 
A note on insurance
If it is not an emergency its a good idea to get in touch with your horses insurance company beforehand if youre thinking that he might need to be put to sleep. They will be able to inform you of any steps they need you to take or your policys requirements such as a second opinion or a postmortem. This way you can prepare yourself should the situation arise.
Following emergency euthanasia the insurance company should also be contacted. They can advise you whether the horse will require a postmortem. Its important to not make arrangements

for the disposal of the horses body until the insurer has been contacted in case they will need the results of a postmortem examination to validate a claim.
Euthanising your precious horse or pony can be a heartwrenching task. Careful research prior to the time and a thorough discussion with your veterinarian or with other horse owners for their experiences can help make this time a little easier.   
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