Geoengineering, in a climate context, means trying to halt or mitigate global warming through means other than reducing greenhouse gas emissions. It comes in two basic flavors:
Taking carbon out of the atmosphere: The first approach would entail sucking carbon dioxide out of the atmosphere. We could do that by planting more trees (although there’s only so much land). We could try to fertilize the ocean with iron to promote the growth of plankton, which absorb carbon dioxide. Or someone could invent a machine to scoop up the carbon directly. None of those are easy.
Blocking the sun: The second type of geoengineering, known as solar radiation management, is far more controversial. This would entail cooling the Earth directly by reducing the amount of solar energy that reaches us. Blocking the sun, in other words. We could put fine particles or liquid droplets into the air to reflect incoming sunlight back into space. Or we could try to increase the reflectivity of clouds by, say, spraying seawater into the atmosphere.
The upside of solar radiation management is that it’s fairly cheap and could potentially avert some of the worst impacts of global warming, such as sea-level rise. See this interview with Harvard climate scientist David Keith, who argues that geoengineering could be a complement to emissions cuts.
The downsides are that this type of geoengineering could have lots of unpredictable side effects, like screwing up global rainfall patterns. Geoengineering would also be difficult to coordinate among various countries — there’d be a great deal of disagreement over who sets the thermostat. And geoengineering does little to address other impacts of rising carbon dioxide emissions, such as ocean acidification.