The Crank-Nicolson method is:
$\frac{u^{n+1}_{i}-u^{n}_{i}}{dt} = \frac{1}{2}(F^{n+1}_{i}+F^{n}_{i})$
This method calculates the next state of the system, i.e. $u^{n+1}_{i}$, by solving an equation involving the previous states and the next state. In the case of the heat equation for example we would get a linear system and if we are using finite elements this system would look like:
$M(\frac{u^{n+1}-u^{n}}{dt}) = -\frac{1}{2}K(u^{n+1}+u^{n})$
or
$(M+\frac{1}{2}dtK)u^{n+1} = (M-\frac{1}{2}dtK)u^{n}$
where M is the mass matrix and K is the stiffness matrix. Because the heat equation is linear we could separate the $u^{n+1}_{i}$'s from the $u^{n}_{i}$'s however the cost of this separation is that we must solve a linear system. In the case of a nonlinear $F^{n+1}_{i}$ and $F^{n}_{i}$ this separation would not be possible and instead we would have to use something like Newtons method to iterate our way to a solution.
More generally, if we have a timestepping scheme of the form:
$\frac{u^{n+1}_{i}-u^{n}_{i}}{dt} = \alpha_{-1}F^{n+1}_{i}+\alpha_{0}F^{n}_{i}+\alpha_{1}F^{n-1}_{i}+\alpha_{2}F^{n-2}_{i}+...$
it is implicit if $\alpha_{-1}$ is non-zero.
Edit:
If you are more familiar with finite differences rather than finite elements then the Crank-Nicolson solution to the heat equation would look like:
$\frac{u^{n+1}-u^{n}}{dt} = \frac{1}{2}A(u^{n+1}+u^{n})$
or
$(I-\frac{1}{2}dtA)u^{n+1} = (I+\frac{1}{2}dtA)u^{n}$
where $I$ is the identity matrix and $A$ is the finite difference tridiagonal discretization matrix.