The basic equation governing the magnetic field due to a current distribution is the Biot-Savart law. Finding the magnetic field resulting from a current distribution involves  a vector product and becomes an inherent calculus problem when the distance from a current to the field point is continuously changing.

According to this law, the magnetic field at a point due to a current element of length dl carrying a current I, at a distance r from the element is

dB = (μ_o/4π) I (dl x r)/r³.

Biot-Savart law has certain similarities as well as difference with Coulomb's law for electrostatic field as there is an angle dependence in Biot-Savart law which is not present in electrostatics.

What is the direction of magnetic field dB due to a current element Idl at a point of distance r from it, when current I passes through a long conductor? 

in the direction perpendicular to both dl and r 

From Biot-Savart law, the magnetic induction is perpendicular to both dl and r, dB = (μ_o/4π) I (dl x r)/r³.