What is a weakness?

While God​ is perfect in every conceivable measure, everything that is not God has some form of shortcoming. These shortcomings can be called "weakness". Some of these may be a general weakness, like actual physical weakness. Others may be a disability of some form, which makes us "weak", even by human standards.

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Why do we boast in them?

According to Scripture, our weaknesses actually glorify God. This can be done in many complex ways. One illustration from our culture might be the Tortoise and the Hare. No one would be surprised if the hare won that race, but it is a memorable story because the tortoise won. In the same way, when God gains victory through a young shepherd boy, that is far more impressive than if He had crushed Goliath with a giant of His own. As Christians, therefore, we can boast in our weaknesses because of the glorious ways our God can use them.