A differential equation is linear if the dependent variable(s) and its derivatives appear linearly in the DEqn. In other words, if the DEqn is a linear combination of $u,u_x,u_y, u_{xy}, u_{xx},...$ then the DEqn will be linear. Here the coefficients of the combination can be nonlinear functions of the independent variables $x,y,...$ in your current notation.
For example,
$uu_x+u_y=0$
is not linear because it has a term with a product of dependent terms $u$ and $u_x$. On the other hand,
$ u_x+x^2u_{yy}+\sin(xy)u_{yy}=0$
is a linear PDE and as you can perhaps see the dependent terms appear linearly.
What is a linear combination? Given $v_1,v_2,...v_k$ a linear combination of the $v_j$'s over $S$ is
$ s_1v_1+s_2v_2+ \dots s_kv_k $
where $s_1,s_2,\dots s_k \in S$. All of this said, the pattern I point out here is merely given to insure that the PDE can be written as a linear operator. I think Robert Miller's answer goes to this point.