They go through different phases. The first phase is prophase, then metaphase, anaphase, telophase, and then they go through cytokinesis and then the new cells start again with interphase.
During Prophase, the nuclear membrane is dissolving and spindle fibers are developing. Chromatin condenses into chromosomes. The microtubules pull the chromatids pairs to the midline of the cell.
During Metaphase, the chromosomes line up at the center of the cell with spindle fibers attached to them. Then, during anaphase, sister chromatids separate and move to oppossite ends of the cell.
During Telophase, chromosomes arrive at the poles and new nuclear membranes form around them. Then comes cytokinesis. For plant cells, a new cell wall forms between the two membranes of the cell plate. But in an animal cell, the plasma membrane pinches inward along the cell's equator until two daughter cells form.
"Oh, I get it now"
After the two new daughter cells form, they will continue with interphase, the between phases of mitosis. And that's how Plant and Animal cells undergo mitosis!