Volcanic eruptions can affect the atmosphere and climate in the immediate vicinity of the volcano and also around the entire planet if the eruption is large enough. The main effect eruptions have on the weather near the volcano is producing large amounts of thunder, lightning and rain.
These effects are due to the large amounts of ash and debris that are released during the eruption, which attract water molecules from the atmosphere. This leads to the formation of clouds and eventually rain. Scientists are still not sure exactly what causes the increase in lightning, but the most accepted theory is that the ash separates into positively and negatively charged particles as it moves through the air.
Eruptions can also lead to large amounts of volcanic fog, as in the case of Hawaii. Volcanoes also release large amounts of sulfur dioxide, which leads to acid rain when it mixes with the water droplets in the atmosphere. This then has the effect of severely diminishing the overall air quality in the area.
Most eruptions are not powerful enough to send ash into the upper layer of the atmosphere, the stratosphere, which is why these effects normally only occur near the eruption. In most cases, the ash only goes as high as the troposphere, where it is washed away due to precipitation. However, larger eruptions that send particles into the stratosphere can lead to either a sudden cooling or warming of the Earth, depending on the size of the particles and how much heat they trap in or sunlight they block out.