By what processes can a volcanic eruption affect temperature around the world

World-wide affects of volcanic eruptions only happens when there are large explosive eruptions that throw material into the stratosphere. If it only gets into the troposphere it gets flushed out by rain. 
the size of the particles (again mostly droplets of sulfuric acid) If they are big then they let sunlight in but don’t let heat radiated from the Earth’s surface out, and the net result is a warmer Earth (the famous Greenhouse effect). If the particles are smaller than about 2 microns then they block some of the incoming energy from the Sun and the Earth cools off a little. That seems to have been the effect of the Pinatubo eruption where about a 1/2 degree of cooling was noticed around the world. Of course that doesn’t just mean that things are cooler, but there are all kinds of effects on the wind circulation and where storms occur. 
even more controversial connection involves whether or not volcanic activity on the East Pacific Rise (a mid-ocean spreading center) can cause warmer water at the surface of the East Pacific, and in that way generate an El Nino. Dr. Dan Walker here at the University of Hawai’i has noticed a strong correlation between seismic activity on the East Pacific Rise (which he presumes indicates an eruption) and El Nino cycles over the past ~25 years. 
volcanism produces about 5X10^11 kg of CO2 per year about 3% of the total CO2 with the other 97% produced by humans. 
hence, volcano have more affect on weather near area it erupts.