In a nutshell
The nucleus is an organelle that stores and protects the DNA in eukaryotic cells.
The nucleus is surrounded by a double-membrane structure known as the nuclear envelope or nuclear membrane. The nuclear envelope separates the contents of the nucleus from the rest of the cell and regulates the movement of substances into and out of the nucleus through nuclear pores.
Within the nucleus, you may also find a small, dense region called the nucleolus. The nucleolus is responsible for assembling ribosomes which in turn is responsible for protein synthesis within the cell.