What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...

The "cyclic cover trick" can refer to more than one thing. One example is as follows. Let $L$ be an invertible sheaf on a smooth projective scheme such that some power $L^{\otimes d}$ has a global section $s$ whose zero scheme $D$ is a smooth Cartier divisor (all of these smoothness conditions are not strictly necessary). Let $\nu:Y\to X$ be the associated degree $d$ branched cover branched over $D$ and whose restriction over $X\setminus D$ is the $\mathbb{\mu}_d$torsor corresponding to $L$. More precisely, $\nu$ is affine and $\nu_*\mathcal{O}_Y$ is the $\mathbb{Z}/d\mathbb{Z}$graded $\mathcal{O}_X$algebra $$\nu_*\mathcal{O}_Y = \mathcal{O}_X \oplus L^\vee \oplus \dots \oplus (L^\vee)^{\otimes (d1)}.$$ Of course we have to say what is the multiplication rule on this algebra. But there is a unique multiplication rule that is $\mathbb{Z}/d\mathbb{Z}$graded and that is compatible with the multiplication rule $s:(L^\vee)^{\otimes d} \to \mathcal{O}_X$. Now, for every $q$, $\nu_*\Omega^p_Y$ has a natural $\mathbb{Z}/d\mathbb{Z}$grading, and the graded pieces turn out to be expressible in terms of tensor products of $\Omega^r_X$ with powers $(L^\vee)^{\otimes s}$. Now assume that $X$ is a $\mathbb{C}$scheme. Then the Hodge theorem gives surjectivity of the various projections $H^r(Y^{\text{an}};\mathbb{C}) \to H^q(Y,\Omega^p_Y)$. Both groups have a $\mathbb{\mu}_d$action that is equivalent to a $\mathbb{Z}/d\mathbb{Z}$grading. In particular, vanishing of certain graded pieces of $H^r(Y^{\text{an}};\mathbb{C})$ will imply vanishing of certain cohomology groups $H^q(X,\Omega^p_X\otimes (L^\vee)^{\otimes s})$. In this way, one can prove the KodairaAkizukiNakano vanishing theorem. So in this case, the "covering trick" is to use a geometric result on the cyclic cover (the Hodge decomposition) combined with computation of the graded pieces to deduce a geometric result on the original scheme. In a similar way, one can prove extensions of the Kodaira vanishing theorem, e.g., the KawamataViehweg vanishing theorem. This is discussed in detail in the book of Kollár and Mori. 

