Excess kurtosis subtracts 3, so a normal distribution has excess kurtosis 0. Positive excess means fatter tails than normal.
The trap. Kurtosis is often described as "peakedness". It is much better understood as tail weight - the fourth-moment weighting means extreme observations dominate the statistic almost entirely.
Consequently it is a terrible estimator in practice: it is extraordinarily sensitive to a handful of outliers, so sample kurtosis from limited data is unstable. For fat-tailed data the population value may not even exist.