Omitted Variable is some "unused variable", which is relevant for the model.
This lack of information will affect the model errors, because information that can be contained in a variable will be contained in $\epsilon$
This side effect can have one very important consequence:
If independent variable $X$ and omitted variable $O$ are correlated, then will be $X$ with $\epsilon$ correlated (because $O$ is the part of $\epsilon$).
We want to predict the salary.
We take the years of education and age in the model.
$$ salary = \beta_0 + \beta_1*{years\_of\_education} + \beta_2*{age} + error $$
Here is obviously many omitted variables, because the salary of the person depends of huge amount of personal qualities and "randomness".
One important omitted variable, that cant be easily measured, is the "Competitiveness" of the person. The Competitiveness affects the salary, but also it affects the education - the most competitive person is more likely to get better grades or do some PhD.
Hence, the education, salary and competitiveness are correlated. Hence are education and error correlated.
Omitted = weggelassene