Conflicting models about how energy transferred in an electric circuit I've been studying DC circuits recently, but I've run into some confusion concerning how batteries deliver energy to circuit elements. 
From what I've read online, it seems that energy in circuits comes from waves that propagate through charges rather than the charges themselves. However, my book seems to convey that charges "pick up" potential energy at a battery and lose it in circuit elements. 
These models of how energy is transferred seem to contradict each other. Which is right? And if the wave theory is right, then what does current or voltage have to do with it? Rather, how do current, voltage, and resistance explain the strength and properties of this wave?
If this matters, I'm using Halliday-Resnick-Krane 5th Edition Physics textbook. 
 A: 
So assuming charge carriers are having their energy reduced through
  circuit elements, what is that energy, exactly? I know the voltage
  drop means there's a dip in potential energy per charge between those
  points, but what is that a energy converted into? 

Like @Aaron Stevens I've never heard anything about waves in DC circuits.  I confess to not knowing anything about non zero poynting vectors that @Alfred Centauri mentioned. 
Some of the potential energy that a battery supplies to electrons is inevitably converted to heat in the resistance that all circuit elements have according to 
$$E_{R}=I^{2}Rt$$
Some of potential energy of the electrons may be converted to energy stored in the magnetic field of any inductance in the circuit, according to
$$E_{L}=\frac{LI^2}{2}$$
And finally some of the potential energy of the electrons may be stored as potential energy in the electric fields of capacitors according to
$$E_{C}=\frac{CV^2}{2}$$

Also, if a circuit has no elements, does the charge not lose any
  charge? That seems intuitive, but I know that goes against the KVL.

At a minimum, there has to be some circuit resistance. If nothing else, all real batteries have internal resistance. If there was no circuit resistance, the charge would accelerate without opposition under the influence of the electric field, resulting in limitless current.
Hope this helps.
