How does it look different from the plasma membrane? We are glad you asked.
This membrane is only found in eukaryotic cells (because only eukaryotic cells have nuclei, right?) and is composed of a double bilayer. This means it has four total layers of phospholipids making up two distinct bilayers. The outer bilayer interacts with the cytoplasm and is physically connected to the rough endoplasmic reticulum (RER). In reality, you can think of the RER as an extension of the outer nuclear bilayer. This close membrane connectivity allows messenger RNA (mRNA) to move directly from the DNA in the nucleus to the ribosomes of the RER without ever coming in contact with the relatively harsh environment of the cytoplasm.
The inner nuclear bilayer is studded with proteins that interact with the contents of the nucleus, especially DNA. Filaments called lamins connect the chromosomes to the inner membrane proteins and help the nucleus keep its shape. The space between the two bilayers is called the perinuclear space. Transcription factors, mRNA, and a few other little guys move across both bilayers and the perinuclear space through large channels called nuclear pores. The most fascinating aspect of these pores is that they can dilate and constrict to allow or block larger molecules’ access across the nuclear membrane,9 much like the pupil in your eye gets larger or smaller to allow more or less light to reach the retina. Crazy stuff.
Lastly, as the cell prepares for reproduction, proteins in the cytoplasm dissolve the nuclear membrane so that the duplicated DNA (something we will dive into a little later) can be separated to opposite sides of the cell. You read correctly; the nuclear membrane dissolves itself. After cell division, new nuclear membranes are formed in both of the "daughter" cells and resume their vital functions in protecting the DNA and providing communication between the nucleus and the rest of the cell.
Nuclei of animal cells actually move around; they do this through cytoskeletal elements like microtubules and actin.