In eukaryotic cells, such as human cells, for example, the cell nucleus is the important organelle that houses the chromosomes. The chromosomes are the store of most of the genetic material of the cell (bar the mitochondrial DNA which is from a separate lineage). The cell nucleus serves the major function of maintaining the integrity of the nuclear genome of the cell. This is the very information necessary for the survival, normal functioning, and ultimately the possible reproduction of the cell itself. It can also regulate gene expression thereby modifying the behaviour of the cell in subtle and important ways.
The chromosomes can be seen as coiled structures that appear in the nucleus and that contain the genes which are used to control protein formation in the cell, as well as histone proteins, and other non-histone proteins that are responsible for the regulation of the DNA. So the cell nucleus can be viewed almost like a command centre for the construction and continued smooth running of the cell. Without it functioning properly everything breaks down. Viruses aim at subverting this command centre by splicing their own genetic material into the host DNA.
Surrounding the entire nucleus is a nuclear envelepe. This structure, with its double membrane, including an inner membrane and an outer membrane, separates the innards of the nuclear structure from the cytoplasm of the cell. The nucleus also contains a structure called the nuclear lamina. This gives physical support to the nucleus. In particular the nuclear lamina provides a mesh on the inner face of the envelope which enables the anchoring of chromosomes and also nuclear pores.
Movement of molecules through the nuclear membrane occurs through nuclear pores because the nuclear membrane is mostly impermeable to molecules. Smaller molecules and ions are allowed through but movement of large molecules like, importantly, proteins must occur through active nuclear transport. This is necessary for both the expression of genes and the maintenance of the chromosomes.
Within the nucleus there also exist several subnuclear bodies. The most important of these is the nucleolus, which is involved in ribosomal assembly and the synthesis of ribosomal RNA. Because it doesn't have a membrane it doesn't count as a true organelle, with some people referring to it as a suborganelle. There are several other non-membranous bodies in the nucleus, such as the Cajal bodies and promyelocytic leukaemia bodies, for example. Such bodies require more research but what is clear is that the nucleoplasm has a variety of functional components in it rather than just being uniform.