What are the differences between implicit and explicit methods when applied to modelling Earth systems?
The answer by @IsopycnalOscillation explains some implications of using explicit and implicit schemes and in what applications one is preferred over the other. Here, I describe how these two methods actually work, and how does the implicit scheme uses knowledge of the later state.

Take the ordinary differential equation:

$$
\dfrac{\partial u}{\partial t} = f(u,t)
$$

and for simplicity, let $f(u,t) = -cu$, where $c$ is a positive constant and $u = u(t)$. The equation is then:

$$
\dfrac{\partial u}{\partial t} = -cu
$$

often called the linear drag equation.

Again, for simplicity, we consider the simplest two schemes, a first order forward Euler differencing (explicit), and a first order Euler backward differencing (implicit). As you know, forward Euler differencing yieds:

$$
u_{n+1}=u_n-c\Delta tu_{n}=u_n(1-c\Delta t)
$$

This method is called explicit because the future state is evaluated as function of the present state. From the von Neumann stability method where the growth factor is defined as:

$$
\lambda = \dfrac{u_{n+1}}{u_n} = 1-c\Delta t,
$$

the scheme is stable when $|\lambda|&lt;1$, neutral when $|\lambda|=1$, and unstable when $|\lambda|&gt;1$. We see that the forward Euler method applied to the linear drag equation may be unstable.

With the implicit Euler backward differencing, we have:

$$
u_{n+1}=u_n-c\Delta t u_{n+1}
$$

Rearranging the terms leads us to:

$$
u_{n+1} = \dfrac{u_n}{1+c\Delta t}
$$

Notice that the $u_{n+1}$ is not found on the right-hand side anymore. Here, the growth factor is:

$$
\lambda = \dfrac{1}{1+c\Delta t}
$$

We see that, in this case, $|\lambda|&lt;1$ holds for any value of $c$ and $\Delta t$.

Thus, if one is concerned about stability, an implicit method is advantageous for solving this equation. However, notice that the implicit method is more computationally heavy, even in this simple case, because a division is significantly more expensive operation than multiplication. This is especially true in larger and more complex systems of equations where $f(u,t)$ may be a large matrix that need to be inverted.

A good introductory text on these methods and there applications in atmosphere and ocean modeling can be found here: Mesinger and Arakawa: Numerical Methods Used in Atmospheric Models (1976). For more references, see the answer to this question.