I found out how to remove a background noise, but then when I export my file in .aiff format and want to copy it in iTunes (in order to make a CD), the noise come back.
First thing, ensure that you are using the current version of Audacity (2.0.5). It is available for free download here: http://audacityteam.org/download/
If that does not fix the problem, you will need to tell us exactly what you are doing (step by step), and possibly post audio samples of the track before and after noise removal. See here for how to post audio samples: https://forum.audacityteam.org/t/how-to-post-an-audio-sample/29851/1
We prefer short audio samples (just a few seconds) in WAV or FLAC format.
You have posted an Audacity project file. It’s just a text file that controls the actual audio in the _data folder of an Audacity project.
Please post an audio file as already requested, preferably WAV, and no more than 1 MB if you attach it here. You can upload larger files to Dropbox or similar and give us the link to download the file. Use File > Export… in Audacity to export a WAV.
If you open Audacity > Preferences, then the Quality section, then set the last drop-down (the second “Dither” box) to “None”, does the noise go away?
There’s something a bit odd here. Looking at the AUP file that you posted, the quietest part of the project is about 20 dB RMS louder than dither noise and the loudest part is 50 dB louder than dither noise (about 300 times louder).