RealClimate has a good introductory post on air capture, which they explain as:
The idea would be to let people emit the carbon dioxide at the source but then capture it directly from the atmosphere at a separate facility.
This is going to be a relatively expensive and complicated strategy for decades — and, of course, you need a place to put the carbon dioxide. That said, a lot of work is going on to see if one can do air capture driven by heat.
Why does that matter? The world has a LOT of zero carbon waste heat not currently being used for anything. Indeed, U.S. thermal power plants alone throw away in waste heat as much energy as Japan uses for every purpose! That’s more than 20 quads. And that doesn’t even count the heat thrown away in industrial processes. Now the smartest thing to do with that heat for the next few decades is obviously either generate electricity with it or use it for heating buildings or industrial processes.
But we should surely do a fair amount of research on air capture, since, by not later than the 2020s, we’re going to get desperate for emissions reductions, and by the 2030s, we’re going to be very desperate and willing to pursue expensive options we that aren’t yet politically realistic.
Indeed, it seems rather likely to me that something like air capture will be needed by the second half of this century. Assuming we actually seriously try to keep emissions below 450 ppm (currently, a doubtful proposition), we’ll probably need to go back to below 400 by 2100 and 350 by 2150, in my optimistic spin on Hansen’s latest paper.
In summary, air capture is not a near-term or medium-term solution, but possibly a long-term strategy.