Convexity stems from the fact that duration is only a good approximation for small changes in yields. For an asset with positive correlation, the estimated price change when yeilds increase will be less negative than out duration model forecasts. Further when interest rtaes drop, bonds with positive convexity will rise by more than a duration model woiuld predict. Thus, for bond holders, duration is a good thing. All non callable bonds have positive convexity.
Negative convexity arises when cash flows are dependent on something the other side can do.
The classic example is a mortgage backed security where if rates move, the borrower can prepay. This prepayment option induces negative convexity.