Biochemical pathways characterize various biochemical reaction schemes that involve a set of species and the manner in which they are connected. Determination of schematics that represent these pathways is an important task in understanding metabolism and signal transduction. Examples of these Pathways are: DNA and protein synthesis, and production of several macro-molecules essential for cell survival. A sustained feedback mechanism arises in gene expression and production of mRNA that lead to protein synthesis if the protein so synthesized serves as a transcription factor and becomes a repressor of the gene expression. The cellular regulations are carried out through biochemical networks consisting of reactions and regulatory proteins. Systems biology is a relatively new area that attempts to describe the biochemical pathways analytically and develop reliable mathematical models for the pathways. A complete understanding of chemical reaction kinetics is prohibitively hard thanks to the nonlinear and highly complex mechanisms that regulate protein formation, but attempting to numerically solve some of the governing differential equations seems to offer significant insight about their biochemical picture. To validate these models, one can perform simple experiments in the lab. This paper introduces fundamental ideas in biochemical signaling and attempts to take first steps into the understanding of biochemical oscillations. Initially, the two-pool model of calcium is used to describe the dynamics behind the oscillations. Later we present some elementary results showing biochemical oscillations arising from solving differential equations of Elowitz and Leibler using MATLAB software.