I've never braided a horses mane before, the one time I was in a horse show way back when I was 12 (I think I was 12) the horses I rode already had the simple hunter type braiding done. Hunter braids are probably the most common braids seen in the show ring but require a shorter mane from my understanding. Not something I would ever want since I love a long flowing mane but the braids do look sharp.
I have seen some horses with braids that haven't been "balled up" and sewn like the ones above. I don't know if there is a technical term but they are simple to do with any length mane to keep the mane tidy and from getting into knots I guess. I think a lot of endurance riders use this so they don't have to fuss with the mane on a long ride. Of course anytime a horse has a braid in they need to be tended to regularly or else the mane can turn into a big knot or the horse could inadvertently rip out a big section of mane while they are itching on a fence post.
With Arabians, Andalusians, Lusitanos or other horses kept with a long mane and tail, the most common approach is the running braid. I think it can look quite elegant but seems a bit tricky to do, kind of like the French braid for women's hairstyles. I doubt I would use it for a trail/endurance horse but to maintain a long mane and keep it nice for the dressage show ring this would be a nice style to perfect.
My all time favorite style of braiding, though it's not a braid at all is the diamond pattern. The first picture shows how to do this, it looks pretty simple, just time consuming but the end result is absolutely stunning!