The carbon cycle describes the balance of many different geologic and biologic processes that move carbon between the atmosphere, the oceans, the earth, and living things.

I think your question is probably about just one of these many processes: photosynthesis, which is the process of plants and algae taking carbon (in the form of carbon dioxide) out of the atmosphere and into the biosphere (living organisms). This process is enabled by the harvesting of energy from sunlight. Photosynthesis happens primarily during the day and also primarily during the summer with the largest number of daylight hours. So this process is cyclical, but it is part of an overall flow of carbon between spheres.

So, it's not that the carbon cycle works differently at night, but the cyclical nature of some processes like photosynthesis are part of the overall balance of the carbon cycle. It also helps to remember that while it's nighttime in one part of the Earth, it's daytime in another. The carbon cycle is on a big picture scale of the entire Earth and atmosphere.

Plants and algae cannot get light during the night, so taking carbon from carbon dioxide uses energy that has to be stored up during the day. For this reason, most plants respire at night just as animals do, but resume photosynthesis during the day. However, some plants do get their carbon at night (mainly because getting carbon for a plant means losing water, and a plant loses less water if it's getting the carbon at night).