Demeaning

Demeaning is the operation $\tilde y_{it} = y_{it} - \bar y_i$ where $\bar y_i$ is the unit-$i$ mean: subtract each unit's average from every observation of that unit. Applied to all variables in a regression, demeaning is algebraically equivalent to adding unit dummies — the Within Estimator view of Fixed Effects estimation.

When to use

Demeaning is the engine inside every fixed-effects estimator. Conceptually it strips out the level of each unit and forces the regression to identify $\hat\beta$ from changes within a unit over time. Any time-invariant regressor (e.g. a state's geographic region) is also demeaned to zero and drops out — which is why FE cannot estimate coefficients on time-invariant variables.