Unmeasured covariate is simply some omitted variable. For example, suppose that the true model that explains wages is given by:
$$w_i = \beta_0 + \beta_1 E_i + \beta_2 A_i + e_i$$
where $w_i$ is wage, $E_i$ your education and $A_i$ innate ability or talent.
Unfortunately in real life we have no statistics about talent/ability of an individual. Consequently in real life we can only estimate the following regression:
$$w_i = \beta_0 + \beta_1 E_i +e_i$$
However, the model above suffers from omitted variable bias (e.g. problem of having unmeasurable covariates see discussion in Verbeek Guide to Modern Econometrics pp 55).
The Diff-in-Diff solves this issue because as explained in the previous question you asked here, even if the outcome is different due to unobserved variable between groups or individuals, you can argue that you can get treatment effect from observing change in trend between treatment and control. This controls for any time invariant unobservable, like for example innate ability or IQ between groups because the differences in these time invariant unobservable is already reflected in difference in outcomes, but it should not affect trend after intervention, since for example if your level of ability grants you \$1000 extra dollars in your monthly wage, it should do so before and after some intervention and so any increase in your wage above what would be given by the trend (for which you use control group) can be argued to be attributed to the intervention.