If I must boast, I will boast of the things that show my weakness. (2 Corinthins 11:30)
You really have to read this whole chapter to understand why Paul talks about boasting. He essentially calls himself a fool, with the understanding that people generally will tolerate what a fool says, because he doesn't know any better. But they still condescend to allow him to speak. So Paul decides to boast as if he were a fool, so that he could lift up God's glory. This is in sharp contrast to the "false teachers" who have followed Paul and tried to get for themselves glory by being "better" apostles than him.
So Paul says that he will boast of things that show his weakness, in order to put the focus on God, who is Paul's strength.
We really don't do that enough, do we? We so often try to build ourselves up and can get totally caught up in "false teachings" just to make ourselves look good. Or even if it's not false teaching, it's a false understanding of our gifts. God did not give us abilities so that we could garner fame and glory, but so that we could glorify Him.
So today let's try something new... let's boast only of God. Let's boast about how He has done amazing things in our lives--despite us. Try it, and let me know how it works out. :)