A tangent, but why exactly are there tiny jewels in them?
There are several different types of jewels used in a watch, the most common are:
Hole Jewels: These are donut shaped jewels that fit over the gear axles (in watch lingo, the wheel arbors).
Cap Jewels: These are flat jewels that are placed on the ends of the axles (arbors).
Pallet Jewels: These are brick shaped jewels on the pallet fork that alternately engage and release the escape wheel. The escape wheel is the gear with funny "boot" shaped teeth.
Roller Jewel: This jewel is on the large balance wheel that swings back and forth. It engages with the pallet fork on the end opposite of the pallet jewels.
Jewels have two important properties that help reduce friction. First, they can be made to be very smooth, and therefore they let the metal parts slide easily. Secondly, they are very hard and therefore don't wear down very quickly. The gears in a watch are carefully designed so that the teeth roll on each other, rather than sliding. If the axle of a gear wears away the hole that it sit in, the gear will shift. That means the teeth will no longer roll on each other and therefore friction will be increased.
The jewels are carefully shaped so that the capillary action of the oil causes the oil to be drawn toward the gear arbors instead of spreading out where it doesn't do any good.
The jewels are located throughout the watch at key spots. The following are typical locations for the jewels:
The basic 7 jewels are part of the escapement and balance and are found on all jeweled watches. They include cap and hole jewels for both the top and the bottom of the balance staff (total of 4), the two pallet jewels and the roller jewel.
The next 8, making 15 jewels, are hole jewels for the fast moving part of the gear train.
The next 2, making 17 jewels, are jewels on the center wheel.
The next 2-4, making 19-21 jewels, are cap jewels on the escape wheel and the pallet fork
Additional jewels are often needed to make automatically wound watches efficiently transfer the small amount of power generated by moving your wrist into the power needed to wind your watch. More jewels are often used for chronograph functions, time repeater chimes, and date/date displays. Very complicated watches can have over 40 functional jewels.
There was also a craze back in the 50's to add many decorative jewels that had no function whatsoever. The "jewel wars" simply took advanatge of the notion that most folks had that the higher the jewel count, the better the watch, which is not always true.
Simply put, jewels act as low friction bearings. To remove them , you really need a staking set, or even better a Seitz jeweling tool.
Jewels may also be set in "chatons." Instead of setting the jewel in a hole drilled in the plate, the jewel is set in a ring (the chaton, often made of gold) and the chaton is screwed to the plate. If this is the case, you will need to remove the chatons first. Also be aware that jewels have little, if any monetary value.
Hope that helps..