The most commonly used bit for a horses mouth is called a snaffle bit. A bit is a snaffle because it creates direct pressure without leverage on the mouth. That means it transmits exactly the same amount of pressure on the mouth as the cowboy is putting on the reins. Other types of bits deliberately amplify the amount of pressure.
There are many different styles but the most common has two pieces of metal which connect in the middle and are hinged so each piece easily moves. At each end there is a oval or D or eggbutt shaped ring typically where the reins connect. The horse feels the bit on several parts of his mouth. The two metal pieces act on his tongue and the side bars. He also feels pressure on his lips and the sides of his mouth.
Many competitions will only permit snaffle bits and the majority of colts are started with a snaffle. The snaffle bit is intended to be used for teaching, controlling, reinforcing, and refining lateral flexion. It is a bit to be used mainly with a sideways pull for lateral flexion. It is not the type of bit you pull on with both hands at the same time, in fact it is highly recommended that you only use one rein at a time with the snaffle.
Interesting, in the English world one definitely does take contact with both hands in a snaffle. I wouldn’t use the word “pull”, but one should never pull. I remember reading a top rider who said, “frank contact, yes, pulling, never!”
Of course in the English world I would say a majority of horses are ridden in a snaffle–there’s not that transition to the curb. If a snaffle is not enough bit then pelhams are relatively common. Personally one of my best horses went in a gag. They have a terrible reputation but he was very happy in it. The girl who bought him from me wanted him for hunters where gags are illegal. She switched him to a pelham and he didn’t seem to go that nicely in it.
Of course you are right Sarah, I wasn’t thinking about the English world.