When current passes through a wire, does it only travel through the outer surface? In electrostatics we are taught that the field inside a conductor is always zero. So when current passes through a wire, the electricity only passes through the surface of the conductor.
In magnetism, we use Ampere's Law to calculate magnetic field inside a wire due to the current passing through each small circular closed loop.
Isn't this contradictory ?
I found many different answers for this web . please clarify .
 A: Your reasoning is faulty.  It is true that in electro-STATICS (ie when there is no movement of electric charges) the electric field inside a conductor is zero.  But this is not the case for electro-DYNAMICS (ie when charges are in motion), eg when there is an electrical current flowing.
It is also true that, for alternating currents, the current becomes more concentrated near the surface of the conductor as the frequency increases (the Skin Effect).  However, when the frequency is low or zero (eg direct current, DC), the current flows throughout the cross-section of the conducting wire, not only near the surface.
I think you are confusing these two different facts.  The first one (no electrostatic field inside a conductor) does not explain or cause the second (alternating current concentrated near surface of conductor).
I don't understand what argument you are making about the use of Ampere's Law and what you find contradictory.
A: When you talk about the current, the electrons enters in the wire from one end and leaves the wire from another end. They walk slowly, however each one pushes the next one in front and you will get an instantaneous current flow (close to speed of light). The ions make a neutralizing background, hence the net charge accumulation during current flow is zero. In normal case there is no electrostatic field on the conducting wire (both inside and outside).
When a charge is accumulated on a conducting surface it tries to be in lowest energy configuration and since similar charge repels each other they settle down to largest possible distance i.e. equally distributed on the surface hence the electric field inside the conductor is zero. 
I hope the explanation about the processes will help you clarify your confusion.
A: 
When you give charge to any sphere like above, charge first spread throughout the conductor and not only surface.
A current flows inside conductor moving electrons and changing electric field in conductor.
Now, positive charge then find that outer surface has less potential than inner. They start moving towards surface in form of current. This current stops when all the charges have reached surface to make potential same throughout the sphere, so that charges find no where to move to.

What did you see?
  
  You saw that current moved throughout inside of sphere to reach at surface. Charges were concentrated at surface only when they stopped moving.
  As a result, electric field at any point inside conductor nullifies.

So, electrostatic is an equilibrium phenomenon.

What happens in wire?
  

Inside wire you apply an electric field. From positive end charges move throughout conductor to reach negative end.
There is no need for them to move towards surface. Electric potential is uniform throughout cross-section of wire. Current is always flowing from one end to other end of wire. Not from center to surface. 
Here , you are not letting an equilibrium situation for charge to be concentrated anywhere (not even at ends).
This dynamic process hence continues.
