Mechanical (or aneroid) barometers use a series of evacuated cells made from metals that will expand or collapse slightly depending on the variations in local atmospheric pressure. While this expansion and contraction is very small in actual motion, levers and catchments (similar to an antique clock) are used to amplify the motion and cause a needle to move against a measuring dial. If not maintained, or left in a humid environment, parts of the mechanism may stick.
Test that the barometer is in fact stuck -- tap the front face. This will usually cause the needle to wiggle a bit; if there's a bit of moisture there, this may solve the problem.
Find the screw for the altitude adjustment for your barometer, and loosen it to see if this increases the apparent sensitivity of the device. If it does not, proceed to the next step. Else, tighten it up again and consult your meteorological station for proper adjustments based on your local conditions and altitude above sea level.
Remove the back case of the barometer with the screwdriver; inside of it, there will be one or more aneroid cells; these will look like coins or flat cylinders. Attached to them will be the casement mechanism, or the tooth and gears by which a small chain will be moved according to a lever attached to the aneroid cells. Using a lint free cloth, wipe down the lever (and make sure it's still free to move), then wipe down the casements.
Apply a lightweight machine oil to all visible gears and chain links by using a few drops at a time from the tip of the oil can. Examine the front of the barometer (and tap the glass) and see if the needle moves. If it does, you've lubricated the part that was stuck. If it does not, examine the back side again and see if there are any levers or gears that do not move freely. Gently move them with your finger to see if corrosion is making them stick, and apply oil as above.
Once the barometer's needle is moving appropriately, reseal the back of the case.
Most watch repair shops can repair aneroid barometers if this does not work.
The technique listed here is for aneroid barometers only. Do not disassemble mercury barometers.