Eukaryote cells generally reproduce themselves using a method called mitosis. When a eukaryote cell senses that there is plenty of food around, it splits into two new cells that together can eat more food. But it’s pretty complicated for eukaryote cells to divide.
They have a lot of DNA molecules, to control all the different parts of a eukaryote cell, like the Golgi bodies and the lysosomes. To make new cells exactly like the old ones, each new cell has to end up with exactly the same DNA as the old cell.
First the DNA molecules build exact copies of themselves out of proteins inside the nucleus. To keep the copies from getting mixed up, each old DNA molecule stays attached to its copy in the middle, making a lot of X-shaped DNA molecules, where one side of the X is a mirror image of the other side.
Then the cell makes two spindles out of protein – bunches of strings of protein. One string of protein from each spindle attaches to each side of each X-shaped DNA molecule. Then the two spindles move to opposite ends of the cell, each pulling their half of the DNA with them and breaking the X apart. At the same time, the nuclear membrane breaks apart (nobody knows exactly how). When the DNA reaches the ends of the cell, it forms new nuclear membranes around it. Now the cell has two nuclei, exactly the same.
Now the cell membrane begins to pinch in half in the middle, being careful to have some mitochondria or some chloroplasts on each side of the middle. Finally the two sides separate and make two new smaller cells. That’s how yeast and other one-celled eukaryotes reproduce, and it’s how your body makes new muscle cells or skin cells when it needs them.