The gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset. In other words, the Gamma shows the option delta's sensitivity to market price changes.
Gamma is important because it shows us how fast our position delta will change as the market price of the underlying asset changes.
Remember: One of the things the delta of an option tells us is effectively how many underlying contracts we are long/short. So, the Gamma is telling us how fast our "effective" underlying position will change.
In other words, Gamma shows how volatile an option is relative to movements in the underlying asset. So, by watching your gamma will let you know how large your delta (position risk) changes.
The above graph shows Gamma vs Underlying price for 3 different strike prices. You can see that Gamma increases as the option moves from being in-the-money reaching its peak when the option is at-the-money. Then as the option moves out-of-the-money the Gamma then decreases.
Note: The Gamma value is the same for calls as for puts. If you are long a call or a put, the gamma will be a positive number. If you are short a call or a put, the gamma will be a negative number.
When you are "long gamma", your position will become "longer" as the price of the underlying asset increases and "shorter" as the underlying price decreases.
Conversely, if you sell options, and are therefore "short gamma", your position will become shorter as the underlying price increases and longer as the underlying decreases.
This is an important distinction to make between being long or short options - both calls and puts. That is, when you are long an option (long gamma) you want the market to move. As the underlying price increases, you become longer, which reinforces your newly long position.
If being "long gamma" means you want movements in the underlying asset, then being "short gamma" means that you do not want the price of the underlying asset to move.
A short gamma position will become shorter as the price of the underlying asset increases. As the market rallies, you are effectively selling more and more of the underlying asset as the delta becomes more negative.
Long Gamma Trading
Take a look at this video from Options Unversity. It provides an overview of the concept of Gamma Trading.