conditional mean and conditional median
In Alecos Papadopoulos's answer, both the conditional mean and the conditional median are linear in $X$. In the following example, the conditional mean is linear in $X$ while the conditional median is quadratic in $X$. This example is a constructed toy but it touches the heart of the issue.

Example: Suppose that $y = \beta_0 + \beta_1 x + u$ and $u = x^2 (w - 1)$, where $w \sim \chi_1^2$ and $x$ and $w$ are mutually independent. Then, as the R command qchisq(0.5,1) gives, $med(w) \doteq 0.455$ so that $med(u|x) = x^2 (-0.545)$, while $E(u|x) = 0$.  We thus have
\begin{equation}
  E(y|x) = \beta_0 + \beta_1 x, ~~\text{ while }~~ med(y|x) = \beta_0 + \beta_1 x - 0.545 x^2.
\end{equation}

Note: Asymmetry of the error distribution and the dependence of $u$ on $x$ are both important. If the error distribution is symmetric, then the conditional mean function and the conditional median function coincide. Also, if the distribution of $u$ is independent of $x$, then $med(u|x)$ is nonzero but constant and thus $med(y|x)$ is again linear in $x$.