What’s the difference between contact discontinuity and shock discontinuity? Is there an intuitive, mathematical way to understand the difference between contact and shock discontinuities?
From the standpoint of hyperbolic PDEs, shocks occur when the characteristic curves of the solution intersect.  But, what about contact discontinuities?
 A: I'll try and explain this in the context of the Euler equations.
As you say, a shock occurs when characteristics (of the same family) converge. Specifically for the Euler equations a shock emerges when two (or more) acoustic characteristics intersect. The simplistic explanation is that the only way to rectify the coexistence of multiple characteristics is a jump discontinuity in the state variables.
The jump discontinuity at a shock is governed by the Rankine-Hugoniot conditions.
Across a shock we see a jump in density, velocity and pressure. We also expect shocks to increase entropy. Furthermore the shock is subsonic wrt to the solution upwind of the shock, and supersonic wrt to the solution downwind.
In addition to shocks, the Rankine-Hugoniot conditions support a second family of discontinuities: contact discontinuities. Similarly, as well as supporting acoustic modes, the Euler equations also support entropy waves. A contact discontinuity is a region of parallel entropy wave characterisics. Across a contact discontinuity, pressure and the normal component of velocity are constant. Density, and entropy can change, but note that the contact discontinuity does not produce entropy itself. There's no requirement for the tangential velocity component to be constant, and indeed it often isn't. The classic example of this would be the Kelvin Helmhotlz instability.
It's probably also worthwhile noting that the linear advection equation only supports contact discontinuities, whereas the inviscid Burger's equation only supports shocks and expansion waves.
A: The classification of discontinuous surfaces itself has nothing to do with characteristics, rather they are based on conservation principles. A clear exposition of the two types of strong discontinuities, namely the shock wave and the tangential discontinuity can be found in $\S$84, Landau & Lifshitz. Take a reference frame fixed to the discontinuity with $x$-axis along the normal. Since mass, momentum and energy is conserved across the discontinuity, we must have for inviscid flows,
$[\rho v_x]=0$
$[\rho v_x^2 + p] = 0, \, [\rho v_x v_y]=0, \, [\rho v_xv_z]=0$
$[\rho v_x(\frac{1}{2}v^2+h)]=0$
where $[m]\equiv m_1-m_2$ denotes the difference between the values of $m$ on the two sides of the discontinuity and the remaining variables take their usual meaning.
In tangential discontinuities, there is no mass flux across the discontinuity, i.e., $\rho_1 v_{1x}=\rho_2 v_{2x}=0$ which implies, provided $\rho_1\neq 0$, $\rho_2\neq 0$, that $v_{1x}=v_{2x}=0$. The $x$-momentum jump then implies $[p]=0$, whereas $y$ and $z$-momentum jumps are identically satisfied with no restrictions on $v_y$ and $v_z$. Similarly, there is no restriction on $[\rho]$. Energy equation is also satisfied identically. Thus, in tangential discontinuities, the density and tangential velocity components can be discontinuities, whereas the pressure must be continuous and the normal velocity component must be zero. Other thermodynamic variables, which can be regarded as a function of $\rho$ and $p$, can also be discontinuous; for example, no constraint is placed on the specific enthalpy $h(\rho,p)$ in the energy equation. A contact discontinuity is a special case of tangential discontinuity in which we assume $[v_y]=[v_z]$=0, i.e., the tangential velocity (and so the velocity) is continuous, but not the density and other thermodynamic variables (of course, except pressure). Since the tangential discontinuities do not have a propagating aspect with respect to the flow, they move with the fluid.
In shock waves, there is a mass flux across the discontinuity, i.e., $\rho_1 v_{1x}=\rho_2 v_{2x}\neq 0$. The $y$ and $z$ momentum then implies $[v_y]=[v_z]=0$., i.e., the tangential velocity is continuous. The normal velocity, the pressure, the density and other thermodynamic variables can be discontinuous and they must satisfy
$[\rho v_x]=0, \,[\rho v_x^2 + p]=0,\, [\frac{1}{2}v_x^2+h]=0$
which is a simplified form the original equations (cancel $\rho v_x$ in the difference formulas and replace $v^2$ with $v_x^2$ because the tangential components being continuous cancel out while taking the difference). Shock waves do propagate with respect to the fluid because of the mass flux in the normal.
The above two discontinuities may be regarded as strong discontinuities opposed to the weak discontinuities in which although the flow variables are continuous, their derivatives may not. We are taking about jumps in the normal derivatives, because by the definition of continuousness across the weak discontinuity, the tangential derivatives are continuous. Weak discontinuities propagate with respect to the gas with the local sound speed. Eg: Boundaries separating a rarefaction wave and a uniform flow.
As we can see the types of discontinuity described so far, obey the jump conditions imposed by the conservation equations. However, we can have what is called an arbitrary discontinuity or an initial discontinuity where conservation equations need not be satisfied ($\S$100 Landau & Lifshitz, $\S$24 Zeldovich &Raizer). An arbitrary discontinuity is prescribed both practically and theoretically, as an initial condition for the problem (initial conditions need not satisfy the equations itself and so they are arbitrary). Of course, such arbitrary discontinuities need not be stable and they will split into the different types of discontinuities mentioned above.
