Interpreting Random v. Fixed-effect Difference-in-Difference equation (+Stata version) When I compute:
xtset panelid year
xtreg y i.treat##i.post i.year

Versus:
xtset panelid year
xtreg y i.treat##i.post i.year, fe

1. What is the actual difference-in-differences (DiD) regression I'm calculating for each? 
2. The coefficient 1.treat should be for the treatment group-- do I need a control group dummy? 
3. In the fixed effect model, 1.treat is omitted due to collinearity, which is to be expected because it is being absorbed by the FE. Correct?
 A: The difference-in-differences equation you are running is
$$
y_{it} = \alpha_i + \gamma \text{post}_t + \beta (treat_i \cdot post_t) + \epsilon_{it}
$$
where $\alpha_i$ are the individual fixed effects (which absorb the treatment dummy), $\text{post}_t$ is the post-treatment time dummy, and $\beta$ estimates your difference-in-differences parameter. When you type in Stata i.treat##i.post then this includes a dummy for i1.treat, i1.post, and their interaction 1.treat#1.post which is equivalent to generating the corresponding dummies by hand and including them - though again the treatment indicator is unnecessary because it is going to be in $\alpha_i$.
You don't need a control dummy because treatment is either 0 or 1, hence a dummy for the control group would be perfectly collinear with the treatment dummy.
Your second specification is correct. Difference-in-differences is essentially a fixed effects regression using the within group variation in the outcomes pre- and post the intervention/treatment. In case you have multiple periods, you can accommodate for this by replace the post dummy with time dummies.
