On average, how many times will I obtain a six by rolling a die N times? If I understand the terms correctly, I am asking: what is the expected value of the random variable defined as the number of sixes obtained by rolling a 6-sided die N times?
I found a source which said that for $N=2$, the expected number of sixes is $12/36=1/3$, but I'm not sure why.
 A: Another way of looking at this problem is that twith two dice there are 36 possible combinations of outcomes.  (1,1, 1,2, 1,3 ...).  If the die is unbiased then each outcome is equally likely.
The 12/26 comes from the fact that there are 11 outcomes that contains sixes, and one of these contains two sixes, so there are 12 sixes in 36 possible results.  
A: Given independence of events (i.e., rolling one die doesn't influence the other roll of the die), the quick calculation is 
$$\text{Expected number of 6's on 2 rolls}=\text{Prob}(x=6)\times N=\frac{1}{6}\times2=\frac{1}{3}$$
A: Another way of stating your answer is to use a mathematical modélisation of the problem.
Since you are repeating $N$ times the experiment "Will I obtain a 6 when rolling a dice", you can model it with a Bernoulli distribution of parameter $p$, the probability that you observe a 6, and $N$ the number of trail.
The expectation of a Binomial random variable is well known and is $pN$, which is in your case $1/3$.
Not that this answer still holds if the dice is biased, but no longer holds if the rolls are not independent.
