There are countless treats which are safe for horses. The majority of fruits and vegetables are safe, along with some foods you might not expect, like candy. In general, as long as you aren't feeding them more than a little at a time, they'll be fine. It's hard to list every single food you could ever consider feeding a horse, so here are the most common ones.
- Hard candy (usually peppermints but also butterscotch and other flavored candies) Don't feed huge amounts of these, since they're so sweet. I usually stick to a maximum of 3 candies at once, and they're best if not fed every day.
- This is a bit political, but current research gives no indication that GMOs and non-organic foods are unsafe for horses
- Broccoli, cauliflower, and other foods which might give your horse gas. There's a slight colic risk.
- Anything from a package, like something you bought frozen or something that isn't obviously a derivative of a fruit. For example, applesauce is safe, but frozen vegetables are risky.
- Anything very sour/spicy; for example, onions
But for the most part, horses can handle a lot more food than you might expect. Just Google ahead of time if you're worried about a particular treat. Also, don't panic too much if your horse accidentally eats something they shouldn't. I've seen horses eat potato chips, ham, and mayo (note that I do NOT recommend any of those) and be totally fine. Of course, if you're concerned, do call the vet.
In terms of the amounts you can feed, I usually limit treats for horses to what I would be willing to eat. For example, I wouldn't eat more than one apple in a sitting, but I might eat a handful of baby carrots at once, so that's all I'll feed. This isn't very precise, but there aren't amounts that're widely accepted for all horses. Since this does vary depending on a horse's size, dietary needs, etc., you should ask your vet if you want precise limits for how many treats you can feed your horse.