What are two other names for a low-pressure system?
what kind of weather are low-pressure systems associated with?

The two other names of a low-pressure system are low depression and cyclone.

The weather associated with low-pressure system is cloudy, rainy or snowy weather.

Explanation:

The low-pressure system is called a cyclone because there is very less atmospheric pressure when compared with the surroundings. These systems help in formation of wind divergence in the troposphere in the upper levels. This process of formation is called cyclogenesis.

To find the type of weather in a low-pressure system we need to observe the "motion of air" in the atmosphere. When "air rises" it cools and this leads to the formation of water vapour. These water vapours condense into liquid and forms water droplets which are otherwise cloudy and precipitated.