In one line
The greeks are four rates of change: how much the premium moves when the index moves, when that sensitivity itself moves, when a day passes, and when the market reprices volatility.
How it works
DELTA is how much the premium changes for a one-point move in the index. A call's delta runs from near zero far out of the money to near one deep in the money, and about one half at the money; a put's runs from zero to minus one. Delta is also, roughly, the market's own arithmetic for the chance that the contract ends in the money — the two are not the same quantity, but they are close enough that traders use delta as shorthand for it.
GAMMA is how fast delta changes. It is largest at the money and close to expiry, which is the mechanical reason a contract's behaviour changes so sharply in its last days: the same index move that barely touched the premium last week moves it a great deal now.
THETA is what a day costs, and it is negative for the holder and positive for the writer. It grows towards expiry rather than running evenly, for the same reason time value does.
VEGA is how much the premium changes when implied volatility changes by one point. It is largest for contracts with time left, and near zero on expiry day — there is no volatility left to reprice.
The desk computes these itself rather than taking them from a data vendor, using the same pricing code the rest of the pipeline uses. That is deliberate: one implementation means the greek on a chart and the greek in a stored row cannot disagree.
What it is not
The greeks say nothing about what comes next, and they are not stable. Each is a rate of change measured at one instant, valid for a small move, and every one of them changes as the index moves and the day passes. A delta of one half is a statement about the premium's sensitivity now, not a claim about where the index goes next.