I'm planning to go from the UK to the States. I was wondering how the ESTA works. Is it all online, meaning when arriving in New York, will they see that I have been approved on the system when they scan my passport? Thus letting me in.
It's one of the most underestimated ESTA questions. Print out your number & attach it to the documents. Despite the fact that automatic registration will give out all the information about you, it may be necessary to have the code in the imprinted form at the airport. Read more here. Print out the PDF with your ESTA number and carry it with your other travel documents. While United States border officials can electronically access your ESTA approval, your airline or carrier may require the printout at check-in.
Yes. It is all electronic, and the information will be available to the border control officers.
It's a good idea to bring a print out of the confirmation page just in case there are any questions, and to show the airline you've completed the ESTA process if necessary.
Admission to the United States is determined by the Customs and Border Protection officers at the border. An ESTA is required to travel to the United States, but CBP will determine you meet the requirements of the visa waiver program before letting you in. An ESTA does not guarantee admission.
You are confusing two concepts, the ESTA and the Visa Waiver Program. The ESTA is an electronic authorization to travel by air or sea to the US. Without that, you can't fly (at least if you're intending to enter the US under the Visa Waiver Program). That's for transportation.
Once you arrive at Immigration, they will (probably) let you in, not because of the ESTA, but because of your passport.