The common Greeks (delta, gamma, vega, and theta) provide an estimate of the sensitivity of an option's price to pricing variables and the associated risk. It's important to note that this is a prediction of option pricing rather than a prediction of what options will make you money.
The importance of the Greeks depends on the level of one's sophistication and the extent of one's option portfolio. Here's my perspective as applicable to what I do on the retail side:
Theta is the rate of change between the option price and time, or time decay. It's important to be aware of it and to understand its non linear behavior but AFAIC, not much more than that.
Gamma, a second derivative, is the rate of change between an option's delta and the underlying asset's price. I have no use for gamma.
Vega represents the the option's sensitivity to volatility and is useful when trading volatility. An example would be the large expected expansion in implied volatility going into an earnings earnings announcement and the large contraction immediately thereafter. It's important for modeling the pricing what ifs of the EA play. Otherwise, day to day, not a significant stat for me.
Delta represents the rate of change between the option's price and a $1 change in the underlying asset's price. This is the most useful Greek for me because when I have a position comprised of 5 to 10 different options, some moving in conjunction with each other and some in opposition, it's important for me to avoid straying too far from delta neutral. To that end, I sell premium on one side or the other to restore neutral.
The Average Joe who executes simple vanilla strategies like covered calls and cash secured puts with the intention of acquisition or sale at a specific price doesn't need to know much, if anything, about the Greeks. Someone at the other end of the spectrum with a large diversified option portfolio would definitely need the Greeks to manage the risk (the extreme would be an options market maker).