what do meteorologists use to measure the climate of a area?

Climate is not the same as weather.

Weather refers to the atmospheric condition at a specific period of time. Sunny, rainy, windy, and cloudy are terms that describe weather. Thermometers, anemometers, psychrometers, rain gauges, and wind vanes are tools to measure weather.

Climate refers to the average weather of an area over a long period of time. "Tropical" and "polar" are terms to describe climate. Average air pressure, humidity, temperature of an area are...

Climate is not the same as weather.

Weather refers to the atmospheric condition at a specific period of time. Sunny, rainy, windy, and cloudy are terms that describe weather. Thermometers, anemometers, psychrometers, rain gauges, and wind vanes are tools to measure weather.

Climate refers to the average weather of an area over a long period of time. "Tropical" and "polar" are terms to describe climate. Average air pressure, humidity, temperature of an area are three key factors used to describe that area's climate. Barometers, hygrometers, and thermometers are the three most common instruments used by climatologists. Barometers are used to measure the air pressure and, therefore, determine an area's average humidity. Hygrometers are used to determine an area's average humidity. Thermometers can be used to find the area's average temperature. Variations of each of the instruments can be read about here, as well as additional tools that are useful to climatologists.