The Wilcoxon test is the non-parametric version of the paired samples t-test - it is for testing whether two groups are different where each participant provides a data point to each of the groups. It differs from the t-test in that it is a non-parametric test; that is, it does not operate on the raw data like the t-test does. Instead, all data are first ranked and the ranks of the two groups are compared. It is used when the data are not normally distributed.
Note: If you are using version 18 of SPSS (also known as PASW 18) there are now two ways of conducting non-parametric tests. To follow the instructions shown in the video follow the same menu selection of Analyse -> Non-parametric Tests and then goto Legacy Dialogs. This will bring up the same menu that you see in the videp.