There are natural (e.g. anaerobic decay or bush fire), industrial (e.g. steel manufacturing) and domestic processes (e.g. cooking) which inevitably will emit carbon into the atmosphere. There are also natural processes which remove carbon from the atmosphere. Plants grow by taking carbon dioxide from air, with sunlight and water. By treating the atmosphere as a big storage, as long as we remove as much as we put in, the storage will remain at the same level.
There are two issues here. First, there is already too much carbon in the atmosphere. Second, we need to stop putting more carbon into the atmosphere.
If we stop putting carbon in the atmosphere and reduce the natural processes of emitting carbon, nature will help us in reducing the carbon in the atmosphere, albeit very slowly. To address the first, we can "offset" the carbon emitted in our industrial and domestic processes by deliberately putting away additional equivalent amount carbon which we extract from atmosphere either artificially or naturally. To address the second, we need to stop land clearing of forest and prevent bush fire and so on.
To put away carbon from atmosphere, first we need to extract the carbon from atmosphere first. While scientists have been working hard, the best way is still to encourage nature to "additionally" remove carbon. Plant growth is one of such method. However, when plant dies, the decaying process will still emit carbon back to the atmosphere. In order to store carbon away, we can use biochar. By storing carbon (as charcoal) in agricultural land, we have a win-win situation of stable long-term storage as well as increase of land agricultural productivity. Since charcoal originates from plant material, if we grow additional plant and convert the plant material into charcoal and store these biochar in farmland, we offset the carbon emission in the other inevitable industrial and domestic processes.
Since we have one atmosphere. The emission in one place can be offset by plant grown in another area with the net effect of creating no additional carbon in the atmosphere. The question becomes what is the most efficient way of achieving this?
To return to per-industrial level of atmospheric carbon dioxide level, those countries which have contributed to the increase in atmospheric carbon dioxide should have the moral obligation to take down those carbon which they have emitted. If the world can agree on a quota, we can have a solution. This is politically unlikely. So, let me just put in a case that every country is obliged to a zero-carbon emission at some point in the future. Since the increase in atmospheric carbon dioxide is from the burning of fossil fuel, we can focus on fossil fuel to start with.
On average we can estimate how much carbon is emitted by measuring the amount of fossil fuels a country consumes. Government can charge a tax (or a price of pollution) on the fossil fuel. The income from the tax can be distributed to the people in the country in an equitable way to offset the inflationary effect of the additional tax. What is that "price" of pollution? Well, that can be determined by a market. Any market consist of buyers and sellers. If a business wants to avoid paying the tax, it can buy offsets. This creates the buyers. People can build business which deliberately put away carbon. These new businesses will become the seller. As long as the tax charged by government is higher than the cost of buying the offset (from where ever in the world), business will minimize the cost by buying the offset. Business will also try to improve their processes in order to minimize the amount of carbon emission in order to remain competitive.
If the government puts an extremely high price to the pollution, there is no doubt many businesses will appear overnight to become the sellers of offset. However, such a measure will be a shock to the economic system. To ease into a trading scheme, government can progressively increase the price at a known rate until a market becomes mature enough to take over. Eventually, polluters should be taxed at a punitive rate and countries should aim at a zero net carbon emission. In the transition phrase, countries should legislate to meet their committed target of reduction as soon as possible.
There are of course many technical details such as how do we ensure that the offset sellers have actually put carbon away for good for a long time. But I hope I have explained the key concept in a carbon trading scheme for laymen.