The Wilcoxon signed ranks test is aimed at testing a null hypothesis from paired replicates data – that both treatments are equivalent. This test is based on one of the two Wilcoxon rank sums as the test statistic.
The Wilcoxon signed ranks test allows you to utilize information about order (rank) of the magnitude of the difference between two replicated measurements. In other words, this test gives more weight to the pairs of measurements which show higher difference. Note that this test does not use the absolute magnitude of the difference, just its rank. In other words, a difference of 4 is considered larger than a difference of 2, but not twice as large, nor 2 units larger.
This is a nonparametric test.