What would be the implications in the Standard Model if we experimentally verified that the electron charge has an intrinsic electric dipole moment? It is known also by recent experiments that the electric charge distribution in space of an electron has been measured with great accuracy to be perfect homogeneous thus essentially geometrically a sphere, meaning within the accuracy limits of the measurement having a zero value of electric dipole moment and therefore the electron charge is a perfect isolated monopole within the accuracy limits of our measurements.
Nevertheless, it is theoretically predicted beyond the current measurement apparatus resolution capability that the electron charge has a very tiny electric dipole moment (EDM) $d_{e}$, thus a very tiny charge inhomogeneity:
$$
U=\mathbf{d}_{\mathrm{e}} \cdot \mathbf{E}
$$
estimated in the order of $10^{-38} \mathrm{e} \cdot \mathrm{cm}$. Also it is predicted that the electric dipole moment is collinear with the direction of the electron's intrinsic magnetic dipole moment (Spin).
Until our measurement methods and apparatus catches up with this degree of accuracy resolution needed there is no way currently to verify this theoretical prediction. Currently our best measurement has a resolution of $1.1 \times 10^{-29} e \mathrm{~cm}$ which means with this upper limit of accuracy the electron charge distribution is a perfect sphere.
My question is, what would be the fundamental implications in the Standard Model (SM) and physics in general if  we finally experimentally verify that the electron has a non-zero electric dipole moment?
I have read the related WP article section about the implications but did not understand. Can you explain them in a more unambiguous way? In what way would this change the current SM status?
 A: The implication would be that there's yet unknown (and not accounted for in the Standard model) source of the $\mathcal{P}$ and $\mathcal{CP}$-violation likely caused by new particles with masses near $\sim 100\,\mathrm{TeV}$.
$\mathcal{P}$ violation means that the natural laws differ when you reflect spatial coordinates. $\mathcal{C}$ violation means that the natural laws are not symmetric under exchange of all particles to their corresponding antiparticles. $\mathcal{CP}$ violation means that if you combine the spatial reflection with exchange of the particles to the antiparticles, this is still not a symmetry. Because of the $\mathcal{CPT}$ theorem, it is equivalent to the natural laws changing with respect to the time reversal.
All these symmetries are violated already in the Standard model but only through a well-known mechanism. There is also no way to add the extra sources of the $\mathcal{CP}$ violation with the particles present in the Standard model. So the discovery of the much larger electron EDM would mean that there is a new physics beyond the Standard model and already within a reach of the next generation of the particle accelerators.
This may partly explain why there are much more baryons (protons and neutrons) than antibaryons in the universe. Partly, because some unknown phase transition in the early universe would also be required to provide the conditions for the disbalance between the baryon-producing and baryon-destroying processes.
While in case of such discovery we will know that there is some new physics beyond the Standard model, we sadly will not know what exactly happens at those energy scales (maybe supersymmetry, maybe something else!) For this we will need new collider, much more powerful than LHC. Nevertheless this will be a very precious piece of information and will likely stimulate the construction of this new collider.
P.S. Don't treat too seriously all this popular talk about shape of the electron.
