Conductors in electric circuits And electrostatics I know from electrostatics that electric field inside a conductor is zero. I also know that there must be electric field inside a conductor carrying current in electric circuits. I am confused. 
 A: By Ohm‘s law the current density in a resistive material is proportional to the electric field in it:
$$\mathbf j=\sigma \mathbf E$$ where $\sigma$ is the conductivity. Since all conductors have a non-zero resistance (except for superconductors), there has to be according to Ohm‘s law an electric field in the conductor to make a current flow.
But why we say sometimes there is no electric field inside a conductor?
Well, this only applies to the static case when there are no currents flowing. Then the charges will distribute in such a way that there is no electric field inside the conductor (this agrees also with Ohm‘s law).
But in a electric circuit we permanently deliver energy through a voltage source. This leads to an ongoing current flow which requires an electric field inside the conductor.
A: 
I know from electrostatics that electric field inside a conductor is zero. I also know that there must be electric field inside a conductor carrying current in electric circuits.

Like the other answer says, you are correct that there must be a field within the conductor for a current to flow because of Ohm's law
$$\vec{J}=\sigma\vec{E}.$$
However, often $\sigma$ is large enough that $\vec{E}$ can be very small. Small enough to not significantly change the solution to a problem relative to assuming the field is 0.
For example, maybe the conductor is a wire connecting to a 1 kohm resistor, and carrying 1 mA of current. Then the voltage dropped across each resistor is 1 V, but the voltage dropped across the wire may be only a few microvolts. In this case we can approximate that the field in the wire as 0, and the voltage drop across the wire as 0 and still get an acceptably accurate solution to our main problem (how much current flows through the resistor).
Or, maybe the conductor is a metal object being probed by a radar beam. If we assume the field in the conductor is 0, we get a perfect reflection from the object. If we consider the finite conductivity of the metal, there is a small absorption of radiation rather than perfect reflection and a tiny phase change in the reflected wave. But these effects may be irrelevant to the main problem of determining whether our radar can detect the object.
