Hartshorne defines a curve over an algebraically closed field $k$ to be a integral separated scheme $X$ of finite type over $k$ of dimension $1$.
This means that locally a curve $X$ looks like the spectrum of some integral finitely generated $k$-algebra.
He then goes on to say that $X$ is projective. What exactly does he mean when he says $X$ is projective and, moreover, how does this affect the local structure of $X$?
He means that $X$ is a projective scheme over $k$; that is, the morphism $X\to\operatorname{Spec} k$ is a projective morphism. This has no effect whatsoever on the local structure of $X$; it is only a global condition. (One precise way of stating this is that any affine curve over $k$ embeds as an open subscheme of some projective curve over $k$.)
Concretely, a projective curve over $k$ is a scheme over $k$ that is isomorphic to a closed $1$-dimensional integral subscheme of $\mathbb{P}^n_k$ for some $n$.