Significant progress has been achieved in leveraging atomic systems for the effective operation of quantum networks, which are essential for secure and long-distance quantum communication protocols. The key elements of such networks are quantum nodes that can store or generate both single and entangled photon pairs. The primary mechanisms leading to the production of single and entangled photon pairs revolve around established techniques such as parametric down-conversion, four-wave mixing, and stimulated Raman scattering. In contrast to solid-state platforms, atomic platforms offer a more controlled approach to the generation of single and entangled photon pairs, owing to the progress made in atom manipulation techniques such as trapping, cooling, and precise excitation schemes facilitated by the use of lasers. This review article delves into the techniques implemented for generating single and entangled photon pairs in atomic platforms, starting with a detailed discussion of the fundamental concepts associated with single and entangled photons and their characterization techniques. The aim is to evaluate the strengths and limitations of these methodologies and offer insights into potential applications. Additionally, the article will review the extent to which these atomic-based systems have been integrated into operational quantum communication networks.