The nucleosome is the basic unit of eukaryotic DNA compaction. It consists of about 147 base pairs wrapped around an octamer of histone proteins. Nucleosomal dynamics provides the availability of packaged DNA for various factors that carry out the vital processes associated with chromatin. It is not completely known how the structure and dynamics of the nucleosome depends on the ionic environment. The current researches do not give an unambiguous answer and often contradict each other. In this paper, we demonstrate supercomputer molecular dynamics simulations of nucleosome models surrounded by monovalent sodium and potassium cations. Analyzing the trajectories, we have shown the details of the distribution of sodium and potassium ions around the linker DNA, nucleosomal DNA at the sites of nucleosomal opening, and histone residues involved in the process of nucleosomal breathing. We have demonstrated the mobility of DNA linkers and the process of nucleosomal unwrapping in various ionic environments, and also assessed the probable mechanisms of the dependence of nucleosome unwrapping on the type of ions in the system. Our study is intended to emphasize the importance of understanding the role of the ionic environment in the functioning of chromatin.