Mechanism

The Sun's energy in the form of electromagnetic waves (radiation) strikes the earth raising our temperature above the near absolute zero of space. Basic physics¹ shows us that in the absence of the atmosphere the earth's surface temperature would average -18C. (frigid, but much better than absolute zero, -273C, of course the absence of an atmosphere might prove fatal to all life on Earth whatever the temperature)

Our atmosphere provides a natural blanket that traps heat by slowing its reradiation back into space. This raises the surface temperature another 28-33C and creates our worldwide average temperature of 10-15C. Ninety percent of the heat trapping is done by water vapor molecules in the atmosphere, but ozone and other impurities including methane and carbon dioxide play a role as well.

And there lies the crux of the problem. If you tinker with the composition of the atmosphere, particularly by increasing the impurities, you change the blanketing effect. Granted, the atmosphere is really big and it takes a huge amount of "tinkering" to make any perceptible difference, but  civilization has been pumping carbon dioxide into the air for centuries. We have actually increased the amount of CO2 in the atmosphere by 35% and we are continuing to add more.  (Keep in mind that CO2 is only a tiny percentage of the atmosphere, just four hundredths of one percent, and even a 35% increase of an impurity is a near negligible change in the overall composition of the atmosphere. Also, keep in mind this impurity consists of 775 x 10¹⁵  grams of carbon, enough to fill 1,500,000 of the largest supertankers.)

We don't really know if a 35% increase in CO₂ is enough to make a difference in the greenhouse blanket effect, but the scientists have been trying to compare the temperature changes they have been observing with the amount of CO₂ in the atmosphere to determine if there is a relationship. Then they try to use that relationship to predict what the temperature would be in the future when even more CO₂ is expected to be present (we are adding a half a percent per year). There are lots of steps and the models are very complex. This is where most of the disagreement is occurring.

The early mathematical models were incomplete and gave answers that could swing from global warming to ice age just by changing an assumption. As time progressed and more effort was put into the models, they improved. The leading models now all seem to be pointing in the same general direction. Some scientists are still skeptical, claiming that scientific "group-think" is at work and contrary opinions are not getting serious attention. The oil companies, particularly Exxon-Mobile have been accused of intentionally fostering dissension. The US government hasn't taken a clear position or acted one way or another.

The global average temperature over the past 800,000 years has varied between 10.5C and 15.5C. (283.6-288.6K). During the past ten thousand years (since stabilizing after the end of the last ice age) temperatures have varied only in a tight 1C band between 14.5-15.5C. In 2006 we are at the top of that range. Under the worst case scenario, the global average temperature will climb another 5.8C by 2100 with catastrophic consequences.

Notes
1. In this case basic does not mean simple. Black body radiation problems require solution of some tricky differential equations. However basic does mean proven. There is no scientific dispute about black body radiation.

About 70% of the Sun's energy that strikes the Earth is absorbed by the atmosphere, land and sea. The rest is reflected back into space. The absorbed energy raises temperatures and is re-reradiated back out into space at a lower frequency typically in the infrared frequencies.  Exactly the same amount of energy must leave as arrives or else the Earth's temperature will change. Anything that upsets this balance can potentially change the temperature.

If the balance is upset the climate will adjust itself to find a new balance. The new balance could be warmer, colder or the same. Sea levels could change. Precipitation patterns, storm intensity, all sorts of things could change. In adjusting to the new balance there may be feedback mechanisms that damp out the effect of the change (negative feedback) or make it more intense (positive feedback).

Carbon dioxide in the atmosphere slows re-radiation by capturing two infrared frequencies commonly emitted by the earth. The surrounding air heats up and radiates at a lower frequency. The radiation eventually escapes the Earth. Radiation is proportional to the 4th power of absolute temperature so the hotter a mass gets the more power it radiates.