Convexity

The convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates.
Duration is a linear measure of how the price of a bond changes in response to interest rate changes.
As interest rates change though the price does not change linearly but is a function of interest rates.
Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes.
Duration can be thought of as the first derivative with respect to interest rates.
The Convexity can be thought of as the second derivative with respect to interest rates.
Convexity is a second order derivative (similar to Gamma for Options)
Convexity can be both positive and negative.

This is the difference between the price/yield relationship (estimated by the modified duration) and the actual relationship
This is a much more complicated calculation
This allows us to predict accurate price movements for changes in interest rates or credit spreads.