This is a slightly adapted version of a reply which appeared on the SPSSX-L mailing list, a mailing list of SPSS users who are largely american-based.
SPSS reports a variety of p-values when using nonparametric tests in SPSS. For example the Mann-Whitney output reports an 'Asymp. Sig. (2-tailed)' and an 'Exact Sig. (2*(1-tailed Sig.))'.
"Asymptotic" means that the p-value approaches the real value as sample size increases. For the Kruskal-Wallis test, the asymptotic value is valid if sample size is above 5 in all groups. For Mann-Whitney's U test, the asymptotic value is valid if both samples involved have sizes above 20 cases.
One problem with the exact p-value is that it is not corrected for ties (several individuals with the same value) so, if you have a lot of "tied" data, the exact p value will under-estimate the real significance (will give higher values that the real ones).