How can a transformer produce a high voltage and a low current? I understand that in ideal transformers, power is conserved. Because of this the product of voltage and current in the secondary winding is a constant.
This means that voltage and current are inversely related, which seems unintuitive because they are directly related by ohms law.
Shouldn't the emf induced in the secondary winding by the alternating magnetic flux be directly related to the current  by some constant, such as the resistance of the secondary winding?
I also came across a term known as impedance that seem to be related to the question. Wondering if it is of any relevance.
 A: 
I understand that in ideal transformers, power is conserved. Because
  of this the product of voltage and current in the secondary winding is
  a constant.

This isn't true.  The expression of power conservation for an ideal transformer is
$$V_s\cdot I_s = V_p\cdot I_p$$
There is no requirement for $V_s\cdot I_s$ to be equal to a constant.

This means that voltage and current are inversely related, which seems
  unintuitive because they are directly related by ohms law.

Power conservation doesn't imply that the secondary voltage and current are inversely related.  Further, the secondary voltage and current (for an ideal transformer) are related by Ohm's law only if the load is a resistor but not otherwise.
For example, when the load is a resistor of resistance $R$ then...


*

*Ohm's Law
$$V_s = R\cdot I_s,\quad V_s\cdot I_s = \frac{V^2_s}{R}$$

*Power conservation
$$V_p \cdot I_p = V_s\cdot I_s$$

*Ideal transformer voltage relation
$$V_s = N\cdot V_p$$


where $N = \frac{N_s}{N_p}$.  Thus
$$I_p = N^2\frac{V_p}{R} = \frac{V_p}{R/N^2}$$
That is, when the load is a resistor, both the primary and secondary voltage and current are related by Ohm's law and power is conserved.  See that the load resistance $R$ connected to the secondary appears as a resistance $R/N^2$ to the circuit connected to the primary.

Shouldn't the emf induced in the secondary winding by the alternating
  magnetic flux be directly related to the current by some constant,
  such as the resistance of the secondary winding?

The resistance of the secondary winding is zero for an ideal transformer.  If the secondary winding has non-zero resistance, power conservation does not hold, i.e., the power delivered to the load is less than the power delivered to the primary.
A: Actually the emf induced per turn in both of the primary and secondary windings are equal due to the conservation of energy. Now if you increase the secondary windings with respect to the primary windings then you will get high voltage than the primary circuit for an ideal transformer. And consequently the current in the secondary windings will decrease following the fact underlies in the constancy of product of current and voltage in bother the windings. If any energy dissipative elements are present in those circuit, then you may convert the primary circuit referred to the secondary or vice versa and apply simply kcl and kvl and rigourously solve the resulting instantaneous equations. If the windings are not coupled to an extent such that magnetic flux leakage has become a serious issue , then the calculation will be quite difficult but not impossible.
A: Here is an analogy which might help you grasp how a transformer works.
Think of an electrical transformer as if it were instead a car transmission. The ratio between torque and RPM at the input shaft is altered by the gears to yield a different ratio of torque to RPM at the output shaft. 
As an example, in first gear the output shaft turns, say, at 1/4th the speed of the input shaft- but the torque at the output shaft is 4x that applied to the input shaft. the transmission "steps up" the torque and "steps down" the RPM, the torque ratio is 1:4 and the RPM ratio is 4:1. 
Now remember that torque is analogous to voltage and RPM is analogous to current. Power is torque x RPM, so since power is conserved in the transmission, torque x RPM at the input shaft = torque x RPM at the output shaft and by analogy voltage x current at the primary winding = voltage x current at the secondary. 
This means our transformer analog then steps current down by 1:4 and steps voltage up by 4:1. 
