What is a barometer and how is it useful for weather forecasting?

A device used for measuring atmospheric pressure.

A barometer measures the force applied by the weight of the column of above the barometer (pressure).

This information is useful for forecasting because there are only 2 ways the atmosphere can actually change, temperature and water content (everything else is a function of these 2 variables). Pressure measures both those variables as warmer air has higher pressure (Boyle's law) and moisture lowers air pressure (N2 is heavier than H2O).