When should we use lag variable in a regression? In some studies, I saw sometimes people used lag of independent variables, sometimes they use lag of outcome variables as an additional control one.
Can I ask what is the mechanic of using lag variable as a control variable and when we should choose lag of an independent variable and when we should choose lag of an outcome variable as another independent variable?

When a lagged explanatory variable is used in a model, this represents a situation where the analyst thinks that the explanatory variable might have a statistical relationship with the response, but they believe that there may be a "lag" in the relationship.  This could occur when the explanatory variable has a causal effect on the response variable, but the causal effect occurs gradually, and manifests in changes to the response later in time.
When a lagged response variable is used in a model, this represents a kind of proxy for auto-correlation in the response variable, and the remaining explanatory variables are then included to see if there is any remaining statistical relationship between these variables and the response, after the effect of auto-correlation are removed.
Both of these situations can occur in a wide variety of econometric settings, since variables in those settings are commonly auto-correlated, and they also often have causal effects on each other that manifest gradually over time.  In terms of when to include these kinds of terms in models, that is a complicated judgment relating to underlying theoretical considerations and diagnostic analysis of the data.  Putting aside theoretical issues, you can look for auto-correlation in regression residuals and you can also look for lagged correlation between explanatory variables and residuals, so this allows you to see if an existing fitted model might benefit from the addition of a lagged model term.
