The nucleus is the most important organelle in the cell.
It contains the genetic material, the DNA, which is responsible for controlling and directing all the activities of the cell.
All the RNAs needed for the cell are synthesised in the nucleus. The nucleolus within the nucleus is the site for RNA synthesis as well as DNA replication.
The ribosomes which are the protein synthesizing machineries of the cell, are made and assembled by the nucleus, in their nucleolus. The ribosomes then enter the cytosol where they either remain free or attach themselves to the Endoplasmic Reticulum and synthesise proteins by translating mRNA.
The lamin proteins that make up the structural framework of the nucleus are directly involved in the initiation of apoptosis (or self induced death) of the whole cell.
A eukaryotic cell cannot survive without a nucleus and dies instantly. (There are exceptions like the erythrocytes which on maturity loses their nucleus).