What important activities occur during each phase of mitosis?

Mitosis is part of the cell cycle, and it is divided into four phases: prophase, metaphase, anaphase, and telophase.  
Prophase:
The chromatin condenses to form visible chromosomes.
The nuclear membrane begins to break down.
The centrioles begin to migrate to opposite sides of the cell.
Metaphase:
The chromosomes (in sister chromatid arrangement) line up across the equator of the cell.
The spindle fibers attach to the centromere of each pair of sister chromatids.
Anaphase:
The spindle fibers begin shortening.
The sister chromatids are pulled apart from each other and migrate to opposite sides of the cell.
Telophase:
The chromosomes begin to uncoil and form chromatin.
The spindle breaks down.
The nuclear membrane forms around the chromatin in what will eventually become the new daughter cells.
At this point, the cell has not fully divided. Cytokinesis occurs next, but it isn't considered to be a phase of mitosis. In animal cells, the cell membrane will "pinch" inward and fully divide the cells. In plant cells, a cell plate forms across the cell's equator to fully divide the cell. 
https://www.ck12.org/biology/mitosis/lesson/Mitosis-and-Cytokinesis-BIO/