What is the tangential component of an electric field is continuous across the boundary between two media? I see the mathematical derivation for the fact that the tangential component of an electric field across two media is continuous, but I don't intuitively understand how this is the case. The electric field should either be impeded or not depending on the material, and this "impedance" for the electric field should affect tangential components as well.
Perhaps, I am misunderstanding something, but I can't intuitively see how this happens.
 A: When you have two dielectric materials with homogeneous $\epsilon_1 , \epsilon_2$ each, applying an external electric field will only produce a bounded charge distribution only on the boundary - the interface between the media.
So locally, you have a plane with charge distribution - thus creating an electric field that is perpendicular to the interface, meaning that the perpendicular electric field changed and has a discontinuity, but the parallel has no reason to change, meaning it is still continuous.    
A: This deduction comes from Faraday's law. Draw a long rectangular loop straddling the interface with width $\epsilon$ and with long edges (of length $L\gg \epsilon$, say) running along the interface. Now apply Faraday's law to the loop. In the electrostatic case, or in the electrodynamic case with finite $\vec{B}$ fields, the magnetic flux $\phi$ and its time derivative through the loop approaches $0$ as $\epsilon$ does so. This will give you the assertion that the tangential electric field on one side of the interface must equal that on the other side.
The same done for Ampère's law shows that the tangential $\vec{H}$ is continuous unless there is a current sheet. So beware in the case of metals: either fully model the current distribution with a continuous tangential $\vec{H}$ assumption (and with a nonzero skin depth), or idealize the current as a sheet but be prepared for a jump in the tangential $\vec{H}$.
The other continuity condition comes from applying Gauss's law to a pillbox with nonzero area ends parallel to the interface and of length $\epsilon$. Unless there is a free charge sheet on the surface, this reasoning leads to the assertion that the normal components of the electric displacement must be continuous. 
So at interfaces between dielectrics of electric constants $\epsilon_1$ and $\epsilon_2$, the electric field lines are continuous, but they have kinks in them (nondifferentiable points). We have 
$$\epsilon_1\,E_{\perp,\,1} = \epsilon_2\,E_{\perp2,\,1}\tag{1}$$
as well as 
$$E_{\parallel,\,1} = E_{\parallel,\,2}\tag{2}$$
which gives you a "Snell-like" law that the electric fields are both in the same plane normal to the interface and:
$$\epsilon_1\,\tan\theta_1 = \epsilon_2\,\tan\theta_2\tag{3}$$
where $\theta_1,\,\theta_2$ are the angles between the electric fields and the unit normals.
This is not Snell's law and it is important to realize that, in the electrodynamic case, one of these electric fields is the sum of both an incident and a reflected electric field, which is why (3) is different from Snell's law. 
