The nucleus is the most obvious organelle in any eucaryotic cell. It is a membrane-bound organelle and is surrounded by a double membrane. It communicates with the surrounding cytosol via numerous nuclear pores.
Within the nucleus is the DNA responsible for providing the cell with its unique characteristics. The DNA is similar in every cell of the body, but depending on the specific cell type, some genes may be turned on or off - that's why a liver cell is different from a muscle cell, and a muscle cell is different from a fat cell. When a cell is dividing, the DNA and surrounding protein condense into chromosomes (see photo) that are visible by microscopy.
The prominent structure in the nucleus is the nucleolus. The nucleolus produces ribosomes, which move out of the nucleus to positions on the rough endoplasmic reticulum where they are critical in protein synthesis.