Keep in mind that delta is dynamic: it changes not only as the underlying stock moves, but as expiration approaches. Gamma is the Greek that determines the amount of that movement.
Gamma is the amount a theoretical option's delta will change for a corresponding one-unit (point) change in the price of the underlying security. In other words, if you look at delta as the speed of your option position, gamma is the acceleration. Gamma is positive when buying options and negative when selling them. Unlike delta, the sign is not affected when trading a call or put.
Just like when you buy a car, you may be attracted to more gamma/acceleration when buying options. If your option has a large gamma, its delta has the ability to approach one hundred (or 1.00) quickly, giving its price one-to-one movement with the stock. Option beginners usually see this as positive, but it can be a double-edged sword. When you have a large gamma, the delta can be affected very quickly, which means so will the option's price. If the stock is moving in your favor, that's great. If it's doing an about-face and moving opposite to your prediction, changes in your option's price may cause a lot of pain.
Gamma is highest for near-term ATM strikes, and slopes off toward ITM, OTM, and far-term strikes. This makes sense if you think it through: an option that's ATM and close to expiration has a high likelihood to accelerate to the finish in either direction.
The graph below shows the gamma for a near-term option (15 days to expiration) with its underlying stock trading around $85. As you can see, gamma is clearly largest for the ATM strike price.
To better explain gamma, we need to revisit delta for a moment. Let's go back to a previous example: an ATM call with a strike price of 40 and the stock also at $40. The new twist is there's only one day remaining to expiration, instead of two months. Delta is still 50 because the option is exactly ATM. If the stock goes up, the call would be in-the-money; if it goes down it'd be out-of-the-money. In other words, you would have a 50/50 chance of the option finishing ITM on expiration. With this in mind, here's an alternate definition and use for the term: delta is a guideline for giving odds of the option finishing ITM at expiration.
Now imagine the stock has moved up to $41, with one day remaining before expiration. The 40 strike call would already be in-the-money. What would the delta be now? Think about the second definition of delta. Being one point in-the-money with only one day remaining means the option has a higher likelihood of staying in-the-money. That likelihood translates into a much larger delta. In this case it may be close to 85.
What would gamma be then? Remember, gamma measures the acceleration factor of delta. When the stock was at $40, delta was 50. When the stock moved up one point to $41, delta increased to 85. The difference between the new delta and the old delta is gamma (85 _ 50 = 35).
If we lengthen the time to expiration in this example, it would drastically change the way the option would act. Let's now say the option has 60 days remaining until expiration, the stock is $41 and the call strike is still 40. What's the probability of the option being ITM at expiration? It's much lower because the stock has more time to move around between now and expiration. To reflect that idea, the delta on this option would be lower _ probably around 60. The gamma was also much lower (around 10 or 0.10 when the stock was at $40).
As with any Greek characteristic, there's a tradeoff to consider. In this case, if you seek out options with high gamma for more acceleration, you're also likely to get high theta (rate of time decay).