So what do you have to do before your travel? You must apply for your ESTA approval, which can be done at this website: https://esta.cbp.dhs.gov/. You will have to fill some of your personal informations like your name, passport number etc. You may also enter some informations about your upcoming journey, but it's not required at the time. In future this may be used as a replacement for I-94W form, which you have to fill, while you are aboard a plane bound for USA.
Once you submit your application you will get your application number, using which you may always review your application process. On the next page you will see a result of your application, which may be one of the following:
- Travel approved - Your application was approved immediately and you can travel to the USA
- Decision pending - Your application will be approved/denied within next 72 hours
- Travel denied - Your request is denied and you can't travel to the USA without a visa, so you must contact a US embassy and apply for a visa
In most cases your travel is approved immediately and you can immediately travel to the USA. If your application is approved, then it is valid for 2 years. It's recommended to apply for ESTA as soon as you start planning your trip. But please note, that there is a requirement of having an electronic passport with biometric informations. If you wish to use a non-electronic passport you will have to get a visa.
The application is free at the time, but the US government is planning to request a 10 USD application fee. But as the European Union is strongly against this fee we all hope, that it will stay free.
You don't have to bring any paper confirmation to the airport with you, because the authorization is stored in a computer system and everything will be controlled automatically when you check in at the airport.
Please note, that you will need ESTA only if you are traveling without a visa. If you have a valid US visa you may use it as you did in past. There's no requirement to have an ESTA while having a valid visa.