This paper presents the study on known algebraic cubic curves and elliptic curves. This paper derives the popular cubic curve or elliptic curve equation from Weierstrass equation from basic scratch, and discusses various cubic curves, elliptic curves and hyperelliptic curves. Various forms of elliptic curves and cubic algebriac curves viz., Ochoa curve, Mordell curve, projection, birational transformation, Conchoid of de Sluze, Maltese cross curve, Semicubical Parabola, Tschirnhausen Cubic, Right Strophoid, Strophoid, are studied.