I change a parameter $F$ in a physics simulation code and then run it, and when the run is finished, I measure a quantity $D$ from the results. For each given value of $F$, the measured $D$ may be different from run to run. So $D$ is a random variable. Theoretically, $D$ is linearly dependent on $F$.
I save the values of $(F,D)$ in a table and perform a simple linear regression analysis with a software like LibreOffice Calc. It produces values for the intercept and the slope of the line along with their standard errors and other quantities. For example, the square of the Pearson correlation ($r^2$) is about 0.99, which, I think, confirms that a linear model is a good choice here.
I'm not sure how I should interpret the standard errors. I think they basically give a measure of how much the intercept and the slope may change if the simulations are repeated (i.e. new samples are produced). In other words, they are estimated standard deviations for the intercept and slope. But what assumptions have been made? For example, what if I repeat the simulations with new set of values for $F$ instead of the same values?
Let $\sigma$ be the standard deviation. Then the standard error is defined as $$SE=\frac{\sigma}{\sqrt{n}},$$ where $n$ is the number of observations in your sample.
So clearly the standard error linearly depends on the standard deviation, but is not quite the same. The more data you gather, the smaller the standard error for given standard deviation.
The standard error is a measure of variance of your estimate (like a mean or an OLS coefficient). For small variance in the data (small $\sigma$) or a lot of independent observations (large $n$), the standard error becomes very small, so you are confident the true parameter value is in a small range around your estimate. This makes sense: As you get more data (larger $n$), idiosyncratic noise will have less and less of an effect on your estimate as it gets averaged out over a lot of independent draws, so the standard error decreases.
I find it very useful to relate standard errors to confidence intervals, which for a given confidence level (say, 95%) tell you in which range the true parameter value likely falls, given the data. The confidence intervals get narrower the smaller the standard error, so you are more confident that the true parameter value is close to your estimate with smaller standard error.