How should these two models be interpreted differently? Specifically, what is the circumstance where you would run one over the other?
aov(Temperature~Day+Error(Subject))
aov(Temperature~Day+Error(Subject/Day))
We'll use an example where I measured the temperature of 10 people once every day for a week. My main interest is to see if the temperature measurements change significantly day-to-day, and I am not interested in the longitudinal trend from Monday to Sunday.
Depending on the contrasts you are using, the R command
aov(Temperature~Day+Error(Subject))
fits a model like
$$y_{ij} = \mu + \beta_j + b_i + \epsilon_{ij},$$
where $y_{ij}$ is the response value for the $i$th individual at the $j$th period (day), $\mu$ is global mean, $\beta_j$ is the effect of $j$th day, $b_i\sim N(0,\sigma_b^2)$ is the Gaussian random effect or random intercept for the $i$th individual and $\epsilon_{ij}\sim N(0,\sigma^2)$ is the Gaussian residual term. The unknown parameters are $(\mu, \beta_j,\sigma_b^2, \sigma^2)$.
On the other hand, the command
aov(Temperature~Day+Error(Subject/Day))
fits the model
$$y_{ijk} = \mu + \beta_j + b_i + b_{ij} + \epsilon_{ijk},$$
where $b_{ij}\sim N(0, \sigma_1^2)$ is a Gaussian random individual-period interaction term. As you can see from the expression, to estimate also $\sigma_1^2$ you need to have replications for each $i$ and $j$, that's the reason for the third index $k$.
