What is $E(X\mid X>c)$ in terms of $P(X>c)$?

$$ \mathrm E(X\mid X\gt c)=\frac{\mathrm E(X\cdot\mathbf 1_{X\gt c})}{\mathrm P(X\gt c)} $$

@Dilip has already provided an example of how to compute $E(X|X>c)$ in the discrete case. In the continuous case, the computation will be:

$$E(X|X>c) = \frac{\int_x x f(x) I(x>c) dx}{\int_x f(x) I(x>c) dx}$$

Thus, if we let $X \sim U[0,1]$ and $c=0.5$ then we have:

$$E(X|X>0.5) = \frac{\int_x x I(x>0.5) dx}{\int_x I(x>0.5) dx} = \frac{(1 - 0.5^2)/2}{0.5}=0.75$$

In terms of interpretation: Intuitively, you can think of $E(X|X>c)$ as the average of all $X$ values subject to each one of them being greater than $c$.