These are some of the tools used in the past (and in many places, still used today) to measure various aspects of weather:
Thermometer: This measures the temperature of air. Most thermometers were glass tubes with liquid such as mercury or alcohol, which quickly expands in response to heat. As the liquid expands it rises in the tube; a corresponding mark outside the tube denotes the temperature. Nowadays thermometers are digital. They contain a resistor known as a thermoresistor, which changes resistance (and thus conduction of electricity) in response to temperature. A simple computer inside the thermometer measures the resistance of the thermoresistor, which it can use to calculate the temperature.
Barometer: A barometer measures air pressure. A rising barometer indicates increasing air pressure, and a falling barometer indicates decreasing air pressure. A high-pressure area is more likely to be dry and clear, while a low-pressure area is more likely to be cloudy and rainy.
Sling Psychrometer: This measures humidity. A sling psychrometer has two thermometers, one without anything on it and another with a wet cloth around it. When one spins the psychrometer around a few times in open air, one causes the water on the cloth to evaporate, which it will do at different rates depending on the moisture in the air. Evaporation results in cooling, which means the thermometer with the cloth will be a lower temperature than the thermometer without the cloth. The difference in temperature on the two thermometers can allow one to calculate the humidity.
Anemometer: An anemometer measures wind speed. Cups on the instrument catch the wind, which turns a dial on the instrument. A corresponding mark on the dial denotes the speed of the wind.
Rain Gauge: A rain gauge allows one to measure the amount of rain that has fallen in a particular time and place.
Wind Vane: A wind vane determines the direction the wind is blowing.