What documents should I take with me to the airport, if I booked my ticket online?

No, you do not need anything other than the printed e-ticket and your passport (which should often be valid for at least six months beyond your travel date to satisfy immigration rules). But there are documents you may have to have. Typically, depending on your passport and destination, to enter a country you may need to apply for a visa before flying. But in any case, this is up to you to figure that, this is not the airline's job.

If you're flying domestically you can board without a passport in most places (still, some identity document may be asked), but a passport will work also.

Finally, printing the e-ticket is not completely necessary in most cases, but it is a good idea for a few reasons, such as reminding yourself of your flight times if your phone battery dies, and convincing immigration officers that you will return home by showing them your return ticket (many immigration areas do not have Wi-Fi).

Since this question is about all documents that you might need, here are some that you might need. But not always.

• Passport, if you are flying to a different country. A national ID will do if your flight is domestic.
• Printout of e ticket. You will need a printed one if you are flying to a different country since you will need to prove your exit date in the next country.

• Printed boarding pass. If your airline allows you to print your boarding pass early, do it. When flying overseas, you will have to go through a document check. But checking-in process is done at this point. I personally don't do this because I collect airliner-printed boarding passes :D

• Make sure you have Visa if the country you are going to has no visa-free or VOA agreement.

• If you plan to work in the other country, employment letter and/or IDs. For conferences, tickets to the conference, etc. Basically, any document to prove your intention if you are going to that country for any reason other than tourist purposes.
• Some airlines ask you to show the credit card used to book the airline. I never had to do this myself, but I have met other people who had to.
• In some countries, you have to fill an entry card before entering the country, the immigration officer keeps half of it, and you have to bring the other half when you exit the country. When you leave the country, try to bring that other half. Airports usually have departure cards, so don't worry too much about them.
• Blank paper, if you are going to Israel. Israel entry/exit seals are a reason to reject your entry to some countries.

Some airlines require you to show the credit card used for the booking at check in. This will usually be noted on your booking confirmation, but it never hurts to check the airline's website to be certain.