Branding puts your brand in front of the horses name, and when you click on the brand it brings you to the stable! So if you look at this horse: https://horsefable.com/horse/1687747 the little teal triangle in front of his name is my brand! I believe you can't unbrand a horse once it is branded, but in my opinion there's no harm in branding. Some people don't like buying horses with brands (there's a forum post around here somewhere about that) but I personally don't care if a horse in branded or not, unless it's multiple emojis. Then I find it to ruin the aesthetic) Hope this helps :) 

To set your brand, go to estate -> manage -> settings. There's 4 characters max and you can use emojis! The brand is always teal, unless its an emoji :)

I brand horses I sell, when i remeber to do so, to prevent others from branding my horses as their own. I find the practise to brand irritating (honestly not many brands loks good and are only disrupting attention from important things). I think it should only be possible to brand horses you breed yourself. To be able to claim someone elses breeding result is disgusting and yes people do so. If someone brand a horse they didn't breed I think at least it should be the brand of the breader that occured.
I tried to set my brand as a "space" but the game wouldn't take it so I had to make a - instead

If a brand is to attention seaking I will not purchase a horse no matter how good it is.

Just like real life, keep track of their horses. 

Also you are probably seeing flowers in manes