The good thing about the conservative form is that this comprises multiple models, such as Shallow Water Equations, Euler Equations or traffic models.
An essential feature of hyperbolic equations is the fact that even continuous initial data $u_0(x)$ can lead to discontinuous solutions, a famous example is (inviscid) Burgers equation with sinusoidal initial data.
This lack of meaningfulness of point data led to the development of Finite Volume Methods, which formulate your scheme in terms of cell/volume averages instead of point values:
$$u_i(t) := \frac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} u(x,t) \mathrm d x.$$
Integration of the conservation law in conservative form $u_t + F(u)_x = 0$ over the $i$'th cell gives
\begin{align}
\int_{x_{i-1/2}}^{x_{i+1/2}} u_t + F(u)_x \mathrm d x = \frac{\mathrm d}{\mathrm dt} \Delta x \cdot u_i(t) + F(u) \Big \vert^{x_{i+1/2}}_{x_{i-1/2}} = 0 \\
\Leftrightarrow \frac{\mathrm d}{\mathrm dt} u_i(t) + \frac{1}{\Delta x} \Bigg( F\bigg(u \Big(x_{i +1/2}^L, t \Big) \bigg) - F\bigg(u \Big(x_{i -1/2}^R, t \Big) \bigg)\Bigg) = 0,\end{align}
irrespective of your particular flux function $F(u)$!
Above, $u\Big(x_{i +1/2}^L,t \Big)$ denotes the value on the left of the cell interface $i \rightarrow i+1$ and $u\Big(x_{i -1/2}^R,t\Big)$ the value on the right of the $i -1 \rightarrow i$ interface. These values have to be somehow determined based on the (only available) cell averages $u_i(t)$. The simplest way would be taking simply the cell average $u_i(t)$, more sophisticated methods use also the neighboring values to construct polynomial interpolations (leading first to degree-one polynomials with limiters and more general to WENO, ENO).
Then the choice of your time integration scheme determines whether you have an explicit scheme or an implicit one, where for general $F(u)$ you would also have to run Newton-Raphson iterations. For FVM, Runge-Kutta methods with a particular property are desired. For an introduction, you can take a look at this, this, this or this paper.
It should be said that in many Finite Volume programs you do not use the flux $F$ coming from your model, but instead a particular approximation (termed numerical flux), leading to the famous Lax-Friedrichs, Rusanov or Engquist-Osher schemes. For the Euler equations in particular, the HLLC scheme is the classical approach.
