Whats the name for.?

Brief：
What is the name for the extra toe in between a horses fetlock and pastern? During evolution, the horse eventually no longer required this toe, but it.. 

What is the name for the extra "toe" in between a horse's fetlock and pastern? During evolution, the horse eventually no longer required this toe, but it is still present on a horse...

Also, is it okay for this apendage to be a bit overgrown?

The people that said "ergot" are correct, the people that said "chestnut", just didn't read the question all the way - the chestnut is higher up on the horses' leg. You can trim them down - they are soft after your horse gets bathed, and often you can pick them off. Another option is to cut or clip them off - you can use hoof nippers, or a large pair of wire cutters (yes, I know it sounds odd, but it works better than scissors). Just be careful not to cut them to close to the skin, because you can make them bleed.

It doesn't hurt anything if the ergot gets long, unless it gets in your horses way, or you are planning to show your horse (makes a neater appearance if you remove it). Ask a vet or something They are called chestnuts, every horse has them, they are actually as unique to each horse as our finger prints are to us, the patterns and shapes of them etc.

A little overgrown is ok, you usually just peel them off when they start to flake, but if they get to long and the horse rips them off they will bleed and can become infected, if your horse has overgrown chestnuts, get out some farriers toe clippers and trim them back, it wont hurt the horse unless you go to skin level.