The simple definition is the difference between the input and output of carbon in a sub system. Therefore, if the output of carbon in greater than the input, then its defined as a carbon source and vice versa for a carbon sink.
The carbon budget has different effects on land, sea and the atmosphere.
On land
An increase in the CO2 in the atmosphere will in the long term stimulate plant growth, leading to an increase in the size of the vegetation cover. Hence, an increase in the sequestration of carbon via the process of photosynthesis. An increase in global temperatures will lead to an increase in the melting of the polar ice caps and tundra in the polar regions, leading to a reduces albedo effect because incoming sunlight will be absorbed by the earth leading to higher temperatures called a positive feedback loop.
On sea
An increase in the level of carbon in the atmosphere will lead to an decrease in the sequestration of carbon hence less acidic waters. Higher temperatures will lead to an increase in the melting of the polar ice caps so more fresh water entering oceans leading to eustatic sea level rises. Also, warmer oceans reduces the diffusion of carbon leading to more carbon in the atmosphere.
On atmosphere
When there is a large store of carbon in the atmosphere, will lead to a reduced albedo effect where the solar radiation will reflect back into the atmosphere trapping the sunlight. As a result, will lead to higher temperatures called global warming. Also, an increase in the level of carbon as stated above will lead to higher temperatures leading to an increase in the rate of evapotranspiration. Therefore, a large transfer of water into the atmosphere where it will cool and condense to form culminus clouds. These will reflect incoming sunlight from the sun leading to lower temperatures on earth called a negative feedback loop.