One of my buddies from school was doing some work at his uncle's house and called me with a question.

Apparently someone had wired the Hot water Heater through a nearby receptacle. The water heater is 208/240 and the receptacle is 120.

My understanding of it is the element will still work, just at half capacity. Is this incorrect?

He is currently correcting it. I just wanted to hear some opinions on the matter.

One-quarter the heat.

Could you throw some theory at me to show how that happens?

Resistance is fixed. Amps is volts divided by resistance. Watts is amps times volts.
Notice volts figures in both equations and one feeds the next so volts gets used twice. Twice the volts is twice the amps and that is again multiplied by volts.
The power formula sums it up with watts equals the square of the volts times the resistance (P=I2R)

At 240 volts and a 24 ohm element:
E = I x R
240 = I x 24
10 = I

W = E X I
W = 240 X 10
W = 2400

At 120 Volts and a 24 ohm element:
E = I x R
120 = I x 24
5 = I

W = E x I
W = 120 x 5
W = 600

Or .5 of the voltage x .5 of the current = .25 of the watts.

Thanks for clearing that up guys.

It makes more sense now that you threw the numbers up.