I've seen two different definitions of an elliptic curve.
1) The first one being that it is a nonsingular projective curve of genus 1.
2) The other definition nonsingular projective curve of dimension 1.
I'm unsure whether these two definitions are the same? In case they are, can someone explain why?
In curves dimension=genus?
Maybe this is wrong, I cannot remember where I saw the "definition" 2.
Definition 2 is wrong. Any curve has dimension 1.
Strictly speaking, definition 1 is also wrong. You need also a base point on the curve to have an elliptic curve; otherwise you just have a genus $1$ curve.
Alternatively, an elliptic curve is an abelian variety of dimension $1$, i.e. an abelian variety which is also a curve. Such a curve necessarily has genus $1$, and is an elliptic curve in the sense of the other definition. The reason for which it must have genus $1$ is essentially because the fundamental group of a topological group is always abelian, and a genus $g$ surface has abelian fundamental group only for $g=0$ and $g=1$; the $g=0$ case is ruled out by the Hairy Ball Theorem (which implies that there is no smooth group structure on the $2$-sphere).