The nucleus of a mammalian cells is approx. 6um in diameter, and makes up 10% of the total volume of a cell.
Inside the nucleus is the nucleoplasm, where nucleotides and enzymes are dissolved (in order to help with DNA replication). It contains the cell's genetic material, formed as a linear DNA molecule.
The main outlines of a nucleus' role is to control gene expression and mediate the replication of DNA.