A monitor bezel is part of an arcade game. This is the part that keeps you from realizing that the screen is just an open frame monitor bolted inside a wooden box.
Many games will have two different bezels. The first is usually an unadorned bezel made from plastic or foam. This one follows the curves of the monitor exactly, and keeps players from being able to see into the machines innards. The second bezel is of a decorative nature. I will describe the different types of them below.
Most decorative monitor bezels are merely a square piece of cardboard (with some sort of picture printed on it), that has an area cut out in the middle, roughly the size of a 19" television screen (may be larger for some games). These are cheaply produced and easily damaged. However, they can often be reprinted inexpensively at your local print shop (to replace the damaged, faded or missing one from your favorite game). Of course you will need a high quality scan to print a new one.
Glass monitor bezels are a whole different animal. They consist of a sheet of rather thick glass, with graphics silk-screened onto the backside of them. These are fairly durable when they are new, but as they age, they become very easy to scratch or damage from the backside (the paint seems to want to lift from the glass). Damaged ones are difficult to repair, and must usually be replaced (and they are expensive, expect to pay at least $50 USD for a glass bezel in good condition, it can be much more). Replacement bezels are made for only the most popular games, and usually run at least $100 USD. (This makes little sense, there have never been any shortage of original Ms. Pac-Man parts, yet they are making new bezels for it. Try and buy a new bezel for Turbo, or Vs. Super Mario Bros., you can't.).
Plexiglass bezels are almost identical to glass bezels, except that they are a bit more durable (my Shark JAWS one is nearing 30 years old and still looks brand new). The paint separation does not seem to be nearly as much of an issue.
Some games have bezel types that are unique. Like the one on Turbo, which has an LED High Score screen imbedded in it. These are very hard to replace in most cases. Try to take care of them.
If you find that your game is missing it's bezel (and you are unable to find a new one), then I recommend ordering a happcontrols generic black bezel, they are both attractive, and inexpensive. You might also want to check eBay, you may be able to find a bezel that is technically incorrect, but still looks right in your game (like my Pac-Man Plus cabinet that has a Pac-Man bezel).