What is the electric field exactly on the surface of a conducting sphere? Within a conducting sphere, the electric field is 0, but is the electric field still 0 exactly on the surface?
 A: The answer is "it depends what you mean by exactly on the surface".
The electric field depends on the amount of charge enclosed. From Poisson's equation:
$$\nabla\cdot E = \frac{\rho}{\epsilon_0}$$
If the charge on the surface is an infinitely thin sheet of charge, then the electric field will be zero on one side of the sheet, and a finite value on the other side of the sheet - with a discontinuity.
In a real conductor, the charge sheet necessarily has a finite thickness (if only because electrons are not infinitesimal); because of this, the electric field will increase continuously from inside to outside this "band" of charge.
A: The point about the E-field being zero inside is a conductor is that you must be writing about electrostatics, the study of charges when they are not moving.
In the ideal world every bit of metal inside your surface (an infinitely thin shell) has no E-field within it.
In the real world when the surface is particular in nature it must be very difficult to decide exactly what is going on.  
I do not think that discussion of an infinitely thin sheet is useful in that it is a totally abstract idea whereas discussion of what happens when you have a one or two or three atom layer of metal probably is because such a sheet can be made.  It is then highly likely that then E-field is not discontinuous as defining the surface is not possible.
A: If you put a charge inside a conducting sphere, you create an electric field inside the sphere. Charges can easily move inside a conductor. The field creates forces, so they will move. 
If the charge is negative, electrons will flow away. They will leave positive nuclei behind. This will cancel the original charge. Likewise, if the charge is positive, electrons will flow toward it until it is cancelled. 
The surface is different. Electrons cannot easily leave the conductor. So they pile up, leaving a negative charge on the surface. Or they get sucked into the interior to cancel a positive charge on the surface. 
Outside the sphere, charges cannot necessarily move so easily. So they cannot flow and cancel the surface charge. There can be an electric field outside the sphere. 
Right at the surface, people usually say the field is discontinuous. It might be more accurate to say it changes abruptly. If you look carefully at how electrons distribute themselves very near the surface, you might find that the electric field penetrates a very short distance.
