Carbon sink means a flow of carbon, which removes carbon from the atmosphere. For example, growing plants sequester carbon from the atmosphere, which means that they are carbon sinks as long as they grow. Carbon sink is not the same thing as carbon storage, which means storing carbon somewhere else than in the atmosphere.
Plants, for example, are a carbon storages, but the size of this strorage may change. When the plants grow, the carbon storage in them increases. This means that they act as carbon sink as well. If the plant does not grow, it still is a carbon storage. If it then starts to decay, it releases carbon to the atmosphere, which means that it acts in an opposite way compared to carbon sink, and then it is called a carbon source and the carbon storage in the plant decreases.