Squeezing shorts is trying to make sure that the shorts (that need to buy back shares) don't find any shares to buy, thus driving up the price (as they'll offer more and more to buy the shares they're obliged to return to those they borrowed them from).

With a gamma squeeze, you also aim to drive up the price, but the mechanism is a different.

As you might know, the payoff of a call option (as a function of the stock price S at expiry) looks something like this:

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The price of a call option before expiry looks basically similar, but smoothed out.

The delta of an option is the first derivative of that graph, or, to put it differently, the change in the price of the option as S goes up. You can convince yourself that when S is very low, the delta of the option is very close to zero, and as S approaches K, the delta increases, and with S much higher than K, delta is basically 1.

(Take an option struck at 50. When S = 10, it's worth basically nothing, and when S = 11, it's still worth basically nothing. So, delta = 0. When S = 200, option is worth about 150, when S = 201, it's worth about 151. So, delta = 1.)

[Gamma is the second derivative. It's basically zero when the stock price is very low or very high, as delta doesn't change. But near K, gamma rises - delta changes!]

Now, if an option trader sells you a call, you're long (you win if the stock goes up), trader is short. Trader doesn't like that risk, so they'll buy the stock to cancel out their price risk. How much stock? Well, depends on delta. If S is small, and delta, say, 0.1, they only need to buy 0.1 stock. Then, for small movements of S (say, a small increase), what they lose on the option, they gain on the stock.

However! If the stock moves substantially, particularly if S approaches K, the delta of the option goes up (gamma is not zero anymore). So, the trader needs to buy more shares to be flat again.

So, here's the strategy: Cheaply buy way-out-of-the-money call options (S << K). Then, manage (somehow), to drive the share price S up towards K. The trader who sold you the options will then buy S, potentially driving up S even further. That's the gamma squeeze.

[Note: The trader that sold you the call charged you a fixed premium at the beginning. Now they must buy the stock when it has gone up, and sell it when it has gone down. So, as the share price goes up and down, they lose money (since they buy expensive and sell cheap). If the share price goes up and down a lot, they'll lose more than they got on the premium. If the share price goes up and down only a bit, say when it moves smoothly in one direction only, or not at all much, then they'll lose less than they got on the premium, so they'll win.

That's why we say that someone that wrote a call is short realised vol: if realised vol is high (higher than the implied one that he charged you for the option), they'll lose, and vv.]

When you wrote an option, you're short vol, short gamma, and long theta. When you bought an option, you're long vol, long gamma, short theta.