Here is an diagram labelling bit parts.
OK, cannons are the
part of the bit that lies on top of the bars. They are measured by their
diameter (how big around they are). Arabs tend to have sharp, thin bars
(per Deb Bennet). Chain bits, twisted wire bits, and the like can be
very hard on the bars as they are slid from side to side. Sawing the
reins on a bit with rough or sharp cannons can be damaging to the bars
(as it cut them and make them bleed).
The cheek is the length of the entire cheek piece. Long cheeks mean extra leverage.
The shank is the length from the cannon to where the rein attaches. Shank length directly translates to leverage.
The purchase is the length from the cannon to where the headstall attaches. A long purchase will cause more poll pressure - it makes the headstall tighten up when the bit is used. Short purchases cannot cause as much pressure.
The link is the length of the middle part of the bit, if it's a 3 piece bit. Long links can be problems for horses with narrow mouths. The link is supposed to sit right over the tongue and work on the tongue and/or palate. However, the attachments for the link can end up right over those thin, narrow bars and dig right into them. I open my horse's mouth and look at where the link rings sit. If they are over the bars or if they look like they will rough up the bars when I pull my horse's head sideways, I know the bit doesn't fit right.