Various types of sufficient and necessary conditions for an analytic set in $\mathbb{C}^n, n \geq 2$ to be algebraic have been proved. In addition to Bishop's paper already mentioned in the comments, there is also the following volume-related criterion:
If $M$ is a pure $k$-dimensional complex analytic subset in the space $\mathbb{C}^n$, $0<k<n$, then $M$ is algebraic if and only if the function $V(r)r^{−2k}$ is bounded, where $V(r)$ is the `area' of the intersection $M_r=M\cap\{z\in C^n:\|z\|\leq r\}$, understood as the integral over $M_r$ of certain canonical differential form.
This was proved in
MR0166393
Stoll, Wilhelm
The growth of the area of a transcendental analytic set. I, II.
Math. Ann. 156 1964 144–170
A different viewpoint has been offered in the papers
MR0219750 Rudin W. A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1968), 671-683 and
MR0294698 Sadullaev, A. A criterion for the algebraicity of analytic sets. (Russian) Funckional. Anal. i Priložen. 6 (1972), no. 1, 85–86.
From these papers, the following criterion arises: An analytic set $M \subset \mathbb{C}^n$ of constant dimension $k$, $0<k<n$, is algebraic if and only if there are vector subspaces $X,Y \subset \mathbb{C}^n$ such that ${\rm dim }X=k$, ${\rm dim Y }=n-k$, $\mathbb{C}^n=X\oplus Y$, and some $C,s >0$ such that $M \subset \{(x,y): x \in X, y \in Y, \|y\| \leq C(1+\|x\|^s)\}$.
Sadullaev also considered algebraicity in terms of growth of certain plurisubharmonic functions. His (and also Rudin's) approach was extended in the following paper:
MR1756571 Zeriahi, Ahmed A criterion of algebraicity for Lelong classes and analytic sets. Acta Math. 184 (2000), no. 1, 113–143
Recently, a volume-based approach to algebraicity of analytic subsets of $\mathbb{C}^n$ was reviwed via the use of amoebas:
MR3393363 Madani, Farid; Nisse, Mounir Analytic varieties with finite volume amoebas are algebraic. J. Reine Angew. Math. 706 (2015), 67–81.