What is a Bit?
A bit is defined by the University of Kentucky's Department of Animal and Food Science as a "tool or means by which the rider communicates [their] wishes to the horse.
Traditionally, bits are made of metals such as stainless steel, sweet iron, and copper. However, bits are not just limited to these materials; they can be made of rubber, plastic, foam, leather, and various other materials.
Bits sit on the bars, a part of the interdental space or the mandible, between the premolars (which are the teeth in the back of a horse's mouth) and the incisors (the teeth in the front of a horse's mouth). This space contains no teeth when proper dental care is in place (Riding Warehouse, n.d.). They also contact the horse's tongue, the muscle that they use to eat and swallow food.
While bits are standard training tools, they are not the sole method available for rider-to-horse communication. Bitless alternatives, such as hackmores and bosals, are also often used to emulate a similar effect of bits.
Sources:
Riding Warehouse. (n.d.). Horse bit & mouth anatomy guide. https://www.ridingwarehouse.com/lc/training/tack/horse-bit-and-mouth-anatomy.html
University of Kentucky Department of Animal and Food Sciences. (n.d.). Bits. Horse Discovery. https://afs.mgcafe.uky.edu/equine/horse-discovery/bits
The Parts of a Bit
While there is a very wide range of the types of bits available, the fundamental structure of a bit can easily be categorized into three components.
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The Mouthpiece
The mouthpiece is present on all bits. This is the central portion of the bit that sits in the horse's mouth, on the tongue and the bars.
Mouthpieces vary significantly in their shape, size and features.
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The Cheekpiece
The cheekpieces are located on the outside edges of a bit. These sit outside the horse's mouth when a bit is in use. They connect the mouthpiece to the bridle components such as the headstall, curb strap, and reins.
Snaffle vs Shank

Snaffles
Contrary to popular belief, a snaffle is not a bit with a single-jointed mouthpiece. In fact, the mouthpiece does not matter at all. A snaffle is defined as a bit with no leverage that exerts direct pressure on a horse's mouth. Various mouthpieces are generally attached to a ring. This ring is also where the headstall and reins are attached.
A snaffle, when engaged, applies pressure to a horse's bars, the corner of their mouth, and their tongue. They are intended to be used without a curb strap, but a curb strap (more commonly called a bit hobble when used with a snaffle) may be attached to the rings of the snaffle below the reins to keep the bit from sliding through a horse's mouth.

Shanks
A shank or curb bit is defined as a bit with leverage that applies pressure to a horse's bars, tongue, corner of the mouth, poll, and chin.
Shanks have two distinct parts, the purchase and the shank. The purchase is the upper part of the shank bit, above where the mouthpiece attaches. The bridle and the curb strap should be attached to the upper ring on the purchase. The shank is the lower part of the bit, below where the mouthpiece is attached; this section is what gives the bit its leverage, and it is where the rein should attach.
A shank must be used with a curb strap to prevent the over-rotation of the mouthpiece within a horse's mouth. This also allows pressure to be applied to a horse's chin.

Gags
A gag is defined as a bit with a sliding mouthpiece. When rein pressure is applied, the mouthpiece physically rises in the horse's mouth while applying downward pressure to the poll. Pressure is also applied to the horses' bars, the corner of the mouth, the chin, and the tongue.
A gag bit should always be ridden with a curb strap to prevent the mouthpiece from over-rotating in a horse's mouth and to prevent overuse of the gag action. This also allows pressure to be applied to a horse's chin.
Gags are controversial bits because they are mechanically contradictory. When a rider pulls on the rein, the gag's cheekpiece slides through the mouthpiece, applying downward pressure on the poll while simultaneously applying upward pressure on the corner of the horse's mouth and its chin. These contradictory pressures can confuse a horse and cause unnecessary stress.
Disclaimer:
It would be my recommendation to avoid using a gag whenever possible. While I do not know you or your horse, the mechanics behind this bit do not lend themselves to productive riding and training.
The Basics of Bit Fit
The fitting of bits is still a relatively new concept; that said, it is best to follow your bit manufacturer's recommendations for how to fit your bit to your horse's mouth.
However, I can offer some general recommendations on what to look for when fitting your bit and how to measure it.
How to Measure a Bit
Bit sizes are generally based on a few basic measurements:
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Mouthpiece length
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Mouthpiece diameter
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Cheekpiece and/or shank length
To measure the mouthpiece length, lay your bit flat and measure from the inside of one cheekpiece to the other. Generally, bits come in sizes from 3" to 7", with the average being around 5".
Measuring the mouthpiece diameter is a little harder; I would recommend going off the manufacturer's measurement if at all possible. If this information is not available, you can take a string and wrap it around the bit. Mark where the string meets and measure the length of the string; this should give you a general estimate of the diameter.
To measure a snaffle ring, measure from the inside of the ring, from one end to the other. This measurement is less important than the other because the bit does not sit directly in the horse's mouth.
Measuring a shanked cheekpiece can be done in two different ways. To measure the entire cheekpiece, measure from the top of the top ring to the bottom of the bottom ring (the outside edges). To measure just the shank portion of a shank bit, measure from the center of the mouthpiece down to the bottom of the bottom ring (the outside edge). These measurements can be a useful reference for determining if a bit is legal by your association's standards.
How to Fit a Bit
How a bit fits your horse can depend on many factors, from tongue thickness to palate shape, jaw space, and dental health. To help find the right bit for your horse, you can take a few measurements.
Measuring your horse's mouth for bit length can be done by putting a wooden dowel or string into your horse's mouth in the general area where the bit sits. Then mark the dowel string at the corner of the horse's mouth. You can then measure the dowel to get a general idea of the bit size you need.
Measuring for bit diameter is not a precise process; you can estimate your horse's plate width by putting your fingers into their mouth on the bars. Starting with one finger, add additional fingers until you feel pressure on either side of your fingers, then measure the width of your fingers to get a general idea of what bit diameter your horse's plate will allow.
Taking measurements of your horse's mouth can be a helpful tool for finding the right bit, but it should not be the only thing you are assessing. It is also important to assess your horse's behavior when the bit is in their mouth. Are they chomping, grinding, or exhibiting pain behaviors? Then it may be time to reassess the fit of your bit.
Another important factor of bit fit is where the bit sits in your horse's mouth. This can be adjusted by lengthening or shortening your headstall. The common saying is that your bit should create about 1-2 wrinkles in the corner of the mouth; however, this can change depending on your bit and your horse's preference.