There are lots of different greenhouse gases, and they vary in how damaging they are. Its quite complex and Im not an expert but greenhouse gases include carbon dioxide, carbon monoxide, CFCs, water vapour, methane, sulphur dioxode, there are many more.
We need some to keep the earth habitable, but too many traps radiation and the environment warms up (thats the theory anyway) which is why we're enocuraged to reduce emissions.
Now some greenhouse gases are more potent than others, and CO2 although abundant is less potent than, say CO (ie cats) and methane (which is why the government will pay you to burn methane, turning it into CO2).
I think we are encouraged to cut CO2 emissions because its one of the things most under our control by, for example, being more energy efficient so less fossil fuel is burned producing electricity.