A lot of techniques assume that data is sampled at regularly-spaced intervals. You might count how much litter is near each mile marker on the highway, or sample points in a forest on a regularly spaced grid (100, 200, 300, ... meters north and 100, 200, 300 meters east of some landmark). This also occurs in time--my EEG machine records a data point every millisecond. We call the interval between adjacent samples the sampling period.
However, a lot of data is not or cannot be sampled with a fixed sampling period. Perhaps the terrain doesn't allow us to place weather stations exactly 50 miles apart. We often study peoples' heights and weights, but these are only opportunistically measured at doctors' appointments (which are often not exactly 1 year apart). These data are irregularly sampled.
The paper you linked describes methods for dealing with the latter kind of data, where the sampling period is not constant. One possible approach is to interpolate your data onto a grid and then use a techniques intended for gridded data. The paper argues that while this works in 1 dimension, it is less satisfactory in multiple dimensions and their lifting-based approach works better.