In the context of options, delta is the change in the option price divided by the change in the price of the underlying stock (i.e., the first derivative w.r.t. the stock price).

Gamma is how much the delta changes for a change in the underlying stock (i.e., the second derivative w.r.t. the stock price).

Market makers hedge their options positions by buying or selling the underlying stock so that they have no exposure to moves in the underlying stock (they are "delta neutral"). So if the delta at the current stock price is 0.50 and the market maker is short 100 call options, he will buy 50 shares. But options are non-linear and the delta changes with the stock price (again, this is what gamma is). If the stock moves up so that the delta increases to 0.60, the market maker will need to buy another 10 shares so that he owns 60 shares and is again delta neutral.

In this way, buying begets more buying and this is what people mean when they talk about a gamma "meltup".