What is an algebraic curve?

Classical approach.

For simplicity, let $\mathbb{K}$ be a field: an algebraic curve $X$ in $\mathbb{A}^n_{\mathbb{K}}$ (the affine $n$-dimensional space over $\mathbb{K}$) is an algebraic set $X$ (the zero locus of a finite family of polynomials with coefficients in $\mathbb{K}$) has pure (Krull) dimension $1$;

what do I mean for "$X$ has pure Krull dimension $1$"? I mean the unique closed, irreducible, proper and non-empty subsets of the irreducible components of $X$ are the points of $X$; equivalently the coordinate ring $\mathbb{K}[X]$ has Krull dimension $1$.

Scheme approach.

Let $X$ be a scheme: it is an algebraic curve if it has pure (Krull) dimension $1$;

for exact, this is equivalent to the existence of an affine open covering $\{\operatorname{Spec}R_i\}_{i\in I}$ of $X$ such that any $\operatorname{Spec}R_i$ has (Krull) dimension $1$; that is the (Krull) dimension of any $R_i$ is $1$. (See Vakil FOAG, December 29 2015 version, definition 11.1.3 and exercise 11.1.B.)