Confusion between Faraday's law and Kirchoff's circuit law
The answer involves vectorial form of KVL from Electromagnetics and the fact that time-varying electric field around a loop formed is non-conservative and hence line integral of electric field intensity which is by definition voltage, is not path independent and not 0 for a closed path, and thus KVL, in its unchanged form, won't hold true...but to hold very useful, KVL still valid, 'mathematical adjustments' were made. This maybe little hard to understand at beginning especially if one is not well versed with electromagnetics theory.

But without going into complex mathematics, I prefer to look at things this way :
Induced emf is by definition, +ve in the direction as given by right hand thumb rule is ( - L di/dt). 

But if you consider inductor as passive circuit element, then it is nothing more than a mathematical adjustment (so that KVL would be applicable in its very well known form) to set voltage drop across it as +L di/dt, the terminal at which current enters being taken positive.

To give simple illustrating example : In a ckt having just an inductor supplied by 1-phase voltage supply,
Induced emf (whose definition states that it is in direction given by right hand thumb rule) in an inductor is 180 degrees out of phase from conventional voltage drop across the inductor, taken positive in the direction of current. I will add figures when I get time. But a look here might be worth : MIT Notes on  inductance 

PS :
This is personally my favorite question (and I'd like to add alot to it later) perhaps because it bugged me until I completed my graduation, but I admit a good answer would involve concerned electromagnetic theory as well.