The most basic panel estimation is the Pooled OLS model,
this model combines all data across indices and performs a regular Ordinary Least Squares Estimation.

A more complex estimation method is the Fixed-Effect (or within) estimator.
If our data only contains to time-periods,
the results of this estimator are equivalent to a OLS estimation of the first-differenced variables.

If our data set contains more than two time periods,
we need to estimate an proper fixed effects model.
This is done by creating a fixed-effect variable for every level along the cross-sectional index (i.e. the non-time index).
A simple way of doing this, is by encoding the cross-section index as a factor and including that factor in the regression (more on factors/categorical variables in the post on Handcoding a Linear Model).