The most rational argument I've commonly heard is the one acknowledging that global warming is occurring, but it's not a "big issue" because it's a natural process. As my art history professor so eloquently said:
"Sure the Earth is warming a little. But, you are arrogant to think that humans could cause this. It is just a part of the natural cycle".
First - even if it was just a natural process, that doesn't automatically disqualify it as an issue of importance to humankind. However, data collected from ice and tree cores, as well as ongoing monitoring of aspects such as temperature, snowfall, rainfall, ocean surface temperature and deep sea currents, help put together a timeline of the earth's climate history.
What we can determine is that though there have been many periods of extreme temperature in earth's past--from ice ages to times when much of the planet existed as a tropical climate, there were always legitimate outside factors that contributed. The biggest of these is the effect of the Sun. The Earth's orbit goes through regular calculable changes which affect how long it is close to the sun, as well as what areas of the Earth are more exposed to it's heating effects. About every 100,000 years the Earth cycles through an elliptical eccentricity of orbit. The more elliptical the Earth's orbit, the more time it spends away from the Sun. Cycles such as these plunge the Earth into predictable temperature swings. There is no such thing as a "natural cycle" of temperature variation, there must be an external factor that's "forcing" the shift. At the moment, none of the historical factors line up. The Earth is not just happening to warm up--we are causing it.
In just the last century or so....
However, we can go farther back than that:
Over the past 100 years, we have been consistently increasing our production of these heat-trapping gases, such as carbon dioxide. Notice the correlation to the temperature index. It is extremely unlikely that the GHG forcing is being caused by anything other than humans. We are simply the only thing that has dramatically changed our output of these gases in this time scale.
Why does this upward trend in temperature matter? Well, at the moment we've been cozy in our coastlines for quite some time, but that might be changing soon
Glacial melt has been increasing dramatically recently, most notably since 2004. Though there is a regular melt-freeze cycle that normally balances the ice loss in the summers, the increase means that the ice can't be replaced fast enough. Additionally, as more ice melts, it triggers factors that only accelerate the process. One of these is the decrease in reflectivity--snow is extremely reflective, which partially shields the glaciers from the sun's energy. However, as the composition of the glaciers is changed by rapid melting, more of this heat is instead being absorbed as the reflectivity decreases. Another is meltwater--as the glaciers melt the runoff drains down and is actually creating a slippery layer between the ice and the land, encouraging parts the fragmenting ice shelf to slide into the ocean. The more these factors occur, the faster the ice melts and moves off the land, which only contributes back to them, creating a positive feedback loop. Because of this loop instead of sea levels rising at a constant pace, the rate is actually accelerating. Scientists estimate that by the year 2100 sea levels will rise as much as 23in, affecting millions of coastal populations.
However, the strain of global warming won't be felt exclusively by these people. Rising temperatures and sea levels will also cause: changes in weather systems and frequency of extreme weather events, decrease in freshwater and water quality, land erosion, flooding from increasing evaporation, out of control wildfires and drought from over-dried land, food scarcity from affected agriculture, not to mention the ruin of the economy from all the damage to the infrastructure and massive loss of jobs and human life....
What can we do about it? Hah...well...it seems like really the best we can do is stop contributing to the speed of the process as much as possible. It's already been set in motion, we can only try to slow it down.