What properties of carbon dioxide make it a greenhouse gas?
According to the Intergovernmental Panel on Climate Change (IPCC):  


  "Greenhouse gases are those that absorb and emit infrared radiation in the wavelength range emitted by Earth."  


In order for a molecule to absorb and emit in the infrared (IR) region, its chemical bonds must rotate and vibrate in a manner that affects something called the molecule's dipole moment. It turns out that due to the symmetry of diatomic molecules like $\ce{O2}$ and $\ce{N2}$, this process cannot happen, and thus these types of molecules cannot absorb in the infrared, which is the wavelength range in which heat is radiated.  

Because $\ce{CO2}$ and other greenhouse gases do not have this kind of symmetry, they can vibrate at specific frequencies within the IR in a manner that affects the molecule's dipole moment, and thus absorb this radiation resulting in the transfer of heat. Much of the heat radiated from the Earth's surface is of the proper wavelength (energy) to be absorbed by these gases, so the criteria given by the IPCC for a greenhouse gas are thus met.  

As I'm not certain of your level of understanding of chemistry, I tried to direct my answer somewhere between purely technical and purely non-technical. If I've left anything unclear to you, please don't hesitate to leave a question in the comments.