Weather "is measured on a scale of hot or cold, wet or dry, calm or storm, or clear of cloudy".(Wikipedia) Weather is generally considered the temperature, windspeed, humidity level, ect of a place. Though it is much more then that. Weather is also storms - be it thunderstorms, tornadoes, or even hurricanes. According to wikipedia weather originates in the atmosphere, therefore so do storms.
Read more at http://en.wikipedia.org/wiki/Weather
Thermometer: A thermometer is an instrument used to measure the temperature of an area or a thing. It is measured in Fahrenheit or sometimes in Celsius. If your measuring liquids, you stick it in the liquid to determine the temp. While if your measuring the temperature outside you.
http://www.centennialofflight.gov/essay/Dictionary/Psychrometer/DI69.htm
Rain Gauge: A rain gauge is an instrument used to measure rainfall. It is measured in inches. The rain gage is a long cylindrical tube that has inches marked on the side so that the user can see how much rain has fallen.
Hydrometer: A hydrometer measures the relative density of liquids. It is unit less.
A hydrometer works by having a weighted glass bulb with a weighted stem. It is placed in the solution then spun. Then the column is read at the waterline to give a number that can be compared to a chart of known values for the particular solution being measured.