Let me give this a try:
1. WHAT IS CONVEXITY
The change can be explained in many ways mathematically, one way is the Taylor Series.
People who uses math in the Financial industry use the term Duration to refer to first order derivative and use the word Convexity to refer to the second order derivative.
Change in Price = -Duration * Delta + 0.5 * Convexity * Delta^2 + ...
In "normal" days, you won't care about the rest of the series as they are negligible and very rarely people care about Convexity even.
It is easy to treat convexity as positive only, but in Finance, there are always two sides, so sometimes convexity can be negative like mortgage backed securities.
(In the US, most home owners can prepay their fixed rate mortgage, like with an embedded call option. When interest rate goes up, the prepayment declines, duration increase, and become more sensitive, when rate goes down, prepayment increases, shortening duration, and less sensitive to decline, sucks both ways)
2. WHY I NEED CONTEXITY
However, when yield curve changes in a non-parallel fashion, things become interesting and high convexity become a safe haven that people pursue as effect is ALWAYS positive. If you have high convexity, hell yeah! You outperform the ones with the same duration when yield goes high or goes low. There is no free meal, for those who knows the yield curve will be volatile but unsure of the direction, convexity is like an insurance that comes with a price. The investors forgive some of the gain and incur losses only when the yield curve stay the same, but should have been any change one way or the other, the insurance pays back.
3. HOW DO I GET CONVEXITY
Bonds with longer duration tend to come with higher convexity, but for the people who try to maintain the same duration, that is where derivatives or options comes in. You can either reduce convexity by selling bonds with embedded options like callable bonds, mortgage backed securities and vice versa. For those who are eligible to buy derivatives without constraint (lots of fixed-income managers are not allowed to touch derivatives), they can purchase future contracts. Future contracts in nature is an EXTREMELY highly leveraged position, the only required investment is the margin to maintain the position.
To give you a sense, a US 2 Year might have a duration close to 2 with an effective convexity of 0.05 while a US 30 year with duration of 22 and convexity of 6 that priced closed to par say $100. However, for a future contract, the price could be only $4 with a convexity of 800 and effective duration of 400!