The keywords for search are "rebase", "merge" and "conflict".
A conflict occurs when different parties make changes to the same document, and the system is unable to reconcile the changes. A user must resolve the conflict by combining the changes, or by selecting one change in favour of the other.
You would find that resolving conflicts (e.g. porting patch to newer version) is usually not trivial operation and cannot be done correctly without deep understanding of code you work with.
Really depends on the change in main code base from mailman-2.1.13 to mailman-2.1.15, but sometimes it is easier to rewrite patch from scratch, sometimes it is sufficient to merge changes from patch to new version and try to fix the conflicts/problems it makes.
I would start with finding out what was the original problem that patch wanted to solve and how it was solved. Then look to 2.1.15 code and find out if the original problem is still there and if it is possible to apply the patch directly (internals didn't change so much in meantime) or new approach must be applied.