A coin mech is the part of a coin operated machine that takes the money (it is technically a "coin mechanism" but no one ever says that, not even the parts catalogs).
There are more kinds of coin mechs than you could ever imagine. I will go over a few of them below.
Tray push mechanism
You see these on pool tables and at the laundromat. You put several coins onto a little tray, and then push the tray into the machine.
Roll down mech
These are common on a wide variety of vending machines. They have a single slot to put your money into, and they have a return slot right below the insert slot (bad coins roll right down to the return, hence the name).
Metal coin mech
These are the most common type. Used in arcade games, vending machines, jukeboxes, etc, etc. They are cheaply constructed, and take only a single kind of coin (usually quarters). These require a separate coin door, but almost any coin door can be mounted up to any metal mech with a little effort.
3 Coin mech
These are the exclusive territory of vending machines. They can take multiple kinds of coins (and can tell the difference between them). These require an external coin slot of some type already installed.
Stealth mechs are solid state mechs that have no moving parts. These are a favorite of stingy operators because they have no way to return change or bad coins. These are also the highest quality coin mechs on the market (average price $80 USD, compared to a mere $15 for a metal mech).