What is the vacuum alignment problem? Can someone explain with a simple example what is meant by vacuum alignment in a field theory? Recently I have heard this term in a seminar and when I asked the speaker I got an unsatisfactory answer: 

vacuum expectation values cannot be arbitrarily chosen.  

But I have often seen in models beyond standard models that additional scalars are assigned zero vacuum expectation value (For example, in two-Higgs inert doublet model, he said that there is no vacuum alignment problem and therefore, one can choose the additional Higgs doublet $H_2$ to have zero vacuum expectation value (vev) i.e., $\langle H_2\rangle=0$). Click here for a review of two-Higgs inert doublet model.
When I searched the internet, I found extremely technical articles related to supersymmetry, technicolor theory etc that I'm not familiar with.
 A: Vacuum alignment is the lifting of the vacuum degeneracy present in SSB, such as chiral symmetry breaking, technicolor, etc,  in comportance with  a small  external perturbation ΔΗ which explicitly breaks the symmetry, simultaneously,
Dashen 1971, Phys. Rev. D 3, 1879  .
SSB only “hides” the symmetry. Recall Goldstone’s 1961 celebrated U(1) sombrero potential,

$$
{\cal L}= \partial \phi ^* \partial  \phi -\lambda (\phi^* \phi -v^2/2)^2 = \tfrac{1}{2}\left ( \partial R \partial R +\frac{R^2}{v^2}\partial \Theta \partial \Theta \right ) -\frac{\lambda}{4}(R^2-v^2)^2,
$$
where we deploy radial variables, $\phi\equiv R e^{i\Theta/v}/\sqrt 2$. 
In these variables, the U(1) symmetry amounts to shifting $\Theta$ and leaving R alone. So the sombrero potential is independent of $\Theta$, and so is its vacuum. Whereas the “Higgs” R must be at the bottom,
$\langle R \rangle =v$, absolutely any v.e.v. for the manifestly massless Goldstone mode $\Theta$ will give the same  energy, since, redefining $R\equiv \rho + v$, 
$$
{\cal L}=  \tfrac{1}{2}( \partial \rho \partial \rho +\partial \Theta \partial \Theta )+ \left
(\frac{\rho^2}{2v^2}   +\frac{\rho}{v}   \right   )\partial \Theta \partial \Theta
-\lambda v^2 \rho^2
-\frac{\lambda}{4}\rho^4 - \lambda v \rho^3,  
$$
so all vacua parameterized by the v.e.v. of $\Theta$ are degenerate, as required.
When, however, one introduces by hand/fiat  a small explicit symmetry breaking mass term $-\Delta H=-m_\Theta ^2\Theta^2/2$ for the goldston,  $m_\Theta \ll m_\rho=v\sqrt{2 \lambda}$, the U(1) shift symmetry is lost. 
This amounts to tilting by a small bit the sombrero off its flat position, with a lowest point at a real minimum of $\Theta=0$, so $\langle \Theta \rangle$ is not arbitrarily chosen anymore, and must be chosen at 0: the vacuum is now unique (non-degenerate, aligned) and the goldstone has turned into a massive pseudogoldston—like the pions of QCD in real life. 
The glorious SSB structure is still around in the infinitesimal $m_\Theta$ limit, but for this “inherent vice” aligning it. A marble at $\Theta=0$ is itching to easily oscillate around 0 at the bottom of the hat, clearly much more easily than up the walls of the hat (corresponding to the $\rho$ excitation—the Higgs). 
Dashen does the heavy lifting of working this out for chiral symmetry breaking in SU(3)×SU(3) in the strong interactions and thereby derives his celebrated eponymous formula for the pseudo-goldstone pseudoscalar masses which outranges our discussion, ($m_\pi^2 f_\pi^2=-\langle 0|[Q_5,[Q_5,\Delta H]]|0\rangle=-\langle 0| \bar{u} u + \bar{d} d|0\rangle (m_u+m_d)/2 ~$ ). 
(A similar formula obtains for pseudogoldstone fermions when susy is broken, instead,  of course.)
