More generally, $x^2-y^3=k$ is called the Mordell equation. For some values of $k$, you'll find elementary solutions in Mordell's Diophantine Equations, and in Uspensky and Heaslett's textbook.
EDIT: I've had a look at Mordell's book. He refers to his paper, The infinity of rational solutions of $y^2=x^3+k$, J London Math Soc 41 (1966) 523-525.
More generally, $x^2-y^3=k$ is called the Mordell equation. For some values of $k$, you'll find elementary solutions in Mordell's Diophantine Equations, and in Uspensky and Heaslett's textbook.