In broad strokes....
Greenhouse Effect
The greenhouse gas effect is a well known process that takes place in our atmoshpere. Certain gases let in sunlight but trap radiant heat, much like the glass in a greenhouse. Without the greenhouse effect, the temperatures on Earth would be very simular to the temperatures on the moon (-387F at night). Likewise with too many greenhouse gases, the earth would have temperatures simular to Venus (900F).
CO2
Many gases contribute to the greenhouse effect to different degrees. Water vapor is the most influential greenhouse gas in our atmosphere. CO2 is also a greenhouse gas, though not the most potent, its abundance makes it influential. CO2 levels in the atmosphere have spiked in the past century or so.
Global Climate & CO2
What is concerning is that data gathered mainly from ice-cores in Antartica show a very close link to global average temperature and C02 atmospheric concentration
Conclusions
With a well known mechanism, and strong historical data, climatologist have created several computer models to predict the effects of rising Co2 in the atmosphere in the next 100 years. Most show a few degrees rise global average temperature, and sea level rise of around a meter.
It is important to remember though, that yearly variation in global temperatures is several degrees and the difference between two years can easily be greater than the total effect of global warming in the next 50-100 years. So attributing a single hot or cold year to CO2 induced climate change is silly.
CO2 induced climate change will slowly shift the global average a few degrees. This will not be that apparent to humans, we see temperature swings of 100F+ in any given year, so nobody will physically notice the difference between a 0-100F year and a 4-104F year. However, the oceans and polar caps will. They will absorb this extra energy in the system causing thermal expansion and glacial melt. This is the most serious effect, all others imho are rather meaningless.