modified-duration draws a straight line through a curve. Convexity measures how much the real relationship bends. For a plain bond the curve bends upward, which is good for the holder: prices rise faster than duration predicts when yields fall, and fall slower when yields rise.
Price change is approximately -duration x dy + 0.5 x convexity x dy^2. Convexity is worth paying for, so high-convexity bonds trade at slightly lower yields. Long zeros and barbells have lots of it; mortgage-backed-security holders are short it.
Example: duration 8, convexity 90, yields fall 1%. Duration alone predicts +8.0%. The convexity term adds 0.5 x 90 x 0.01^2 = 0.45%, so the real gain is about 8.45%. On a 1% selloff the loss is only about 7.55%.
Related: modified-duration, negative-convexity, duration, bond-price-yield-relationship