You're talking about cheek pieces. Hundreds of mouth pieces can go with these.
Mouth pieces vary based on your needs:
Loose ring: swivels in the horses mouth, allowing the bit to sit in a more natural position in the mouth
d-ring: basically standard operating procedure, keeps the bit in one place while allowing the reins to move more freely
Full-cheek: for horses that toss their head, it keeps the bit from sliding through the mouth. Please use bit keepers when using a full cheek, as these put the bit in the correct mouth position.
I use an egg but snaffle with my mare, and she is CRAZY!I used to use a loose ring with her but then I bought an egg but to try it out and it works great!I think loose rings work very well.But it has happened to me with D rings where it gets stuck then my horse starts freaking out.Hopefully this helped!Good luck finding a bit!
Many people just use a snaffle bit and I do too. This is just my preference. I really don't have any problems with a D-ring bit but I don't use loose rings so I don't know about that. My recommendation is trying out different bits and seeing what your horse likes. Horses' can be sensitive about bits if they don't like it. I hope this helped! :) ( I also think a snaffle looks best with a figure eight)