Most people have been through lectures on bitting that although informative sometimes need a bit of a reality check. The vast majority of horses go well in a snaffle and that's all they'll ever need but then there are those that need a bit of thought when it comes to bitting.
The next step up from a snaffle is usually something with a bit of poll pressure like a gag. It's also worth noting when you use a three ring gag on the lowest hole that the sides can sometimes turn up leaving you with no brakes. The solution seems to be to put a strap on the underside in the curb groove that goes between the two largest rings on the bit (use a spur strap). However, check the rules before adding to bits as "home made" changes are sometimes not allowed.
Looking at more specialist cases. The waterford is a useful tool for horses that clamp down on the bit and run. It's made of large metal bubbles so doesn't give the horse anything to lean or clamp down on.
Although quite funky looking the Tom Thumb is also quite a handy bit and works well for those horses that you can never seem to find the right bit for. Usually because it's such an unusual bit that no one's ever tried it before.
The world of the double bridle is more for the dressage side of things but the same principles go. Keep it simple. So stick with the standard bridoon and weymouth unless you have to change things. Looking at port sizes is also a good plan if your horse has a chunky tongue... Double bridles can be used XC or showjumping but you do need to be careful not to get your knitting in a knot. Pelham roundings can help save the hassle of two sets of reins if you can't quite get your head round it.
Hackamores are again a specialised bit of kit but have kept more than one top end horse going in the past when they've had mouth injuries. Some horses also just go better without a bit in their mouth. Hackamores need a fair bit of attention to get the right fit and type though as they can exert some pretty harsh pressure if used incorrectly. The bottom line being that the strongest bit in the right hands can be very kind but the softest bit in the wrong hands can be lethal.
There are lots of other types of bits and people classify bits in many different ways but that covers what most horses go in. It's also worth noting the usefullness of the Fulmer cheek. It's not only useful for young horses that have steering issues but has been used for several ''cart horses in a breastplate'' who need that bit more encouragement to go round the corners. Before I land myself in it, several top top horses have been ridden in Fulmer cheeks so it's not just for babies and lunies. Happy bitting!!