Free resolutions are central objects in commutative algebra with many applications in algebraic geometry. In this talk, we provide an introduction to the properties of the minimal graded free resolution of the coordinate ring of an embedded projective variety, the shape of which gives rise to the notion of "syzygies". We will see how these algebraic quantities relate to the geometry of the embedded variety and discuss the recent developments in the study of syzygies of embedded algebraic curves and higher dimensional varieties.