What's the difference between fixed effect regression and (panel) fixed effect regression?
Fixed effects regression is one of a broader class of panel econometric methods which take advantage of panel data to difference out unobserved variables which would cause biased estimators in a naive ordinary least squares regression. It is sometimes also called the "within" estimator since it is based on deviations of individual observations from group means. So, they are the same thing.
The Wikipedia page provides a good overview: https://en.wikipedia.org/wiki/Fixed_effects_model.