What is the most important part of cells and lies at the center of all cellular activity?
Usually the nucleus is given the honor of being called the most important part of the cell. This is because the nucleus holds all of the DNA in a cell, and the DNA contains all of the information necessary for life processes. In many analogies involving cell organelles, the nucleus holds the place of the 'brains' of the cell.
Although every cell with a nucleus hold all of the DNA necessary for the living organism, not all DNA is used by every cell. When a cell differentiates, or becomes a certain type of cell, it uses only the DNA necessary for that specific cell. In an analogy I often use with students, I refer to the nucleus as holding the entire instruction book for life, but cells only 'read' the chapters that apply to them.
This is the idea behind some of the research that is trying to make cells become different types of cells. If we can figure out how to get a cell to 'read' other parts of their DNA, they can become a different type of cell. This can help lead to new treatments and/or cures for a wide variety of conditions.
To answer this question, it's important to understand that there are two basic categories of biological cells: eukaryotic cells include all plant and animal cells and contain membrane-bound organelles and a discrete nucleus in which DNA is stored; and prokaryotic cells, by contrast, typically belong to bacterial species and are devoid of membrane-bound organelles or a discrete nucleus.
In eukaryotic cells, the nucleus is considered the center of all cellular activity. The primary purpose of the nucleus is to facilitate the storage and transcription of DNA, which provides the genetic instructions for the synthesis of proteins. Within the nucleus, DNA is packed tightly into molecular formations called chromosomes, composed of long strands of DNA wrapped around proteins called chromatids. Specialized enzymes in the nucleus unravel the DNA from chromosomes and transfer its stored information to RNA, a related molecule whose purpose is to provide a template for protein synthesis in locations throughout the cell. The RNA is transported to ribosomes, which are present both in the cytoplasm and on the surface of internal membranes such as the nuclear envelope and the endoplasmic reticulum; in the ribosomes, the unique sequence of genetic information stored in the RNA is used to encode the synthesis of a particular protein. The nucleus plays a central role in this process, which is collectively termed DNA transcription and translation.
Prokaryotic cells still contain DNA, but it is not stored within a membrane-bound nucleus; instead, it clusters in a region of the cell called the "nucleoid" (or "nucleus-like") region. DNA transcription and translation still occurs in these cells, but it does so without the additional protection and efficiency that the nucleus provides to eukaryotic cells.