A centrosome is a cellular structure involved in the process of cell division. Before cell division, the centrosome duplicates and then, as division begins, the two centrosomes move to opposite ends of the cell. Proteins called microtubules assemble into a spindle between the two centrosomes and help separate the replicated chromosomes into the daughter cells.
The centrosome is an important part of how the cell organizes the cell division. There are a lot of processes that need to be coordinated together when you have two cells, both their nucleus and the cytoplasm, moving away from each other. Microtubules create a spindle, and that's really the structural elements of the cell that coordinate the cells moving away from each other. And the centrosomes organize the microtubules, so it's called the microtubules organizing center. The centrosomes duplicate before cell division, so they then help to organize the microtubules and the cell division process.