Bionanotechnology has revolutionized nanomaterial synthesis by providing a green synthetic platform using biological systems. Among such biological systems, microalgae have tremendous potential to take up metal ions and produce nanoparticles by a detoxification process. The present study explores the intracellular and extracellular biogenic syntheses of silver nanoparticles (SNPs) using the unicellular green microalga Scenedesmus sp. Biosynthesized SNPs were characterized by AAS, UV-Vis spectroscopy, TEM, XRD, FTIR, DLS, and TGA studies and finally checked for antibacterial activity. Intracellular nanoparticle biosynthesis was initiated by a high rate of $Ag^+$ ion accumulation in the microalgal biomass and subsequent formation of spherical crystalline SNPs (average size, 15-20 nm) due to the biochemical reduction of $Ag^+$ ions. The synthesized nanoparticles were intracellular, as confirmed by the UV-Vis spectra of the outside medium. Furthermore, extracellular synthesis using boiled extract showed the formation of well scattered, highly stable, spherical SNPs with an average size of 5-10 nm. The size and morphology of the nanoparticles were confirmed by TEM. The crystalline nature of the SNPs was evident from the diffraction peaks of XRD and bright circular ring pattern of SAED. FTIR and UV-Vis spectra showed that biomolecules, proteins and peptides, are mainly responsible for the formation and stabilization of SNPs. Furthermore, the synthesized nanoparticles exhibited high antimicrobial activity against pathogenic gram-negative and gram-positive bacteria. Use of such a microalgal system provides a simple, cost-effective alternative template for the biosynthesis of nanomaterials in a large-scale system that could be of great use in biomedical applications.