Mitosis

Often used as a synonym for cell division, mitosis is actually the division of the nucleus. Along iwth cytokinesis, mitosis makes up the M-Phase of the cell cycle (Interphase being the other part of the cycle. The nucleus is arguably the most important organelle in the cell because it contains the DNA that codes for the proteins that determine everything the cell is and does.

Copied DNA must not be damaged, but it must have space to separate, so it can't stay in the nucleus
Each new cell must have one, and only one, copy of the DNA, so the process must be organized
Once the DNA is separated and protected in two distinct nuclei, the cell can proceed with division (cytokinesis)

The M-Phase

Mitosis - dividing the nucleus so that the new cells can each have the right identical DNA.

Prophase - the chromosomes condense (think spaghetti twisted into macaroni) so that it can be easily separated with minimal damage. The nuclear membrane also disintegrates so that chromosomes can spread out.
Metaphase - Sister Chromatids (exact copies of the same chromosome) line up in the middle of the cell and spindle fibers attach each copy to the opposite end of the cell.
Anaphase - Copies are pulled apart to opposite sides of the cell.
Telophase - Nuclear membranes form around each group of chromosomes making two separate nuclei.

Cytokinesis - a ring of contractile proteins around the circumference of the cell (imagine a belt around the waist pulled way too tight) contracts until the cell splits into two separate daughter cells.