Zero filling is the process of extending an FID with extra points of zero intensity. After fourier transformation the extra points interpolate between the experimental points smoothing the spectrum and often increasing resolution.
In the figure below an FID that was acquired using 16K complex points is transformed with no zero filling (black), zero filling to 16K (red), and zero filling to 32K (blue). Note that after fourier transformation only the real data is displayed so acquisition of 16K complex data points gives a spectrum containing 8K real data points.
The multiplet in the figure shows an expansion of the spectrum where the effect of zero filling can be easily seen. The black points and solid line show the spectrum transformed without zero filling. Doubling the number of data points and extending the real data to 16K ("zero filling once"), results in a new red point being interpolated between every black experimental point. Doubling the data points again to 32K gives three blue interpolated points between every black experimental point. Notice how the points in the zero filled spectra overlap exactly with the corresponding experimental data points; zero filling has simply added extra points between the experimental points.
Looking now at the lines that join the points, particularly at the center of the multiplet, one can see how the resolution improves with zero filling. The solid black line is the spectrum without zero filling, while the dotted blue line is the spectrum that has been zero filled four times. Note too, how the peak maxima also shift slightly.
If you need to accurately measure peak positions, or are looking for maximum resolution, additional zero filling is a good idea, particularly in 1D spectra. For multidimensional spectra excess zero filling rapidly increases the size of the processed data and can make the files unmanageable so should be used judiciously. For 1D spectra though, trying additional zero filling may allow you to extract more information from your experiment. In general, the standard processing parameters in the Facility are set up to zero fill once.