To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?
This book has six chapters, and one needs perhaps different prerequisites for each chapter. For chapter $I$ and $II$, on geometry of cubic curves and points of finite order, just basic algebra is enough, together with plane curves (affine and projective). For elliptic curves over the complex numbers, a bit of complex analysis is required. For chapter $III$ and $IV$ on the group of rational points and cubic curves over finite fields, more algebra is needed (for example, for the proof of Mordell's theorem). For chapters $V$ and $VI$ on Integer points on cubic curves, and Complex multiplication, Galois representations, algebraic number theory, and algebraic geometry would be helpful.