All visitors to the USA, including infants and children must have their own valid permit for entering the country. If you are travelling for business or leisure this will usually mean having a valid ESTA (Electronic System for Travel Authorization). This is essentially the USA’s authorisation to travel under the Visa Waiver Program (VWP) whereby citizens of certain countries, including the UK, are permitted to travel to the USA without having to apply to the US Embassy for a visa.

Winter in New York City

The ESTA is valid for 2 years from the date of approval or until your passport expires, whichever is soonest. It is applied for online and approval is usually instant.  It can however take up to 72 hours so try not to leave it until the last minute if you can. You don’t need to print off the approval and carry it with you since it is associated with your passport number, although it may be worth printing it off for your records. I’ve never printed it but I do usually make a note of the authorisation number.

Unfortunately there is a fee for the service which must be paid for each individual application. Although the ESTA website is vague about the current fee (I couldn’t find it anywhere,) it is actually $14.00 per application so does add up for families. Do beware of websites that advertise ESTA but charge more than $14.00. The
only official website is the one below, you cannot get an ESTA from any other source. All the information you need as well as details on eligibility are here, you can also apply or check on an current application or see if a previous ESTA approval is still valid. 
The official Department of Homeland Security page

National Mall, Washington, D.C.

When you arrive in the US you will be asked to present a completed Customs Declaration Form. You will either be given this by the airline when you check in or on the aircraft before you land. Make sure you complete it before arriving in the immigration hall as this will speed up your progress. You will need information such as the name and address of the hotel or friends you will be staying with to complete your form. You will be heavily questioned if you don’t know where you’re heading so it’s worth having it written down somewhere convenient. For more information visit the Customs and Border Protection website