iTunes gives the option to convert mp3s into "lossless" formats such as AIFF and Apple Lossless. This is probably a stupid question but is the conversion actually supposed to improve sound quality?

No: lossless = mp3 unchanged = no quality drop or gain.

Of course there will never be a quality gain if you convert any file to any other type of file. The quality will be always be lower or the same. The same quality will only happen if you convert to a lossless file. And quality loss will always happen when you convert to a lossy file.

The Apple Lossless encoder will exactly reproduce the MP3 audio you feed into it, flaws and all. GIGO. (Garbage in, garbage out)

NEVER. You should compress CD source material (WAV) with lossless.

it only get bigger.

Does the iTunes music store sell lossless files?

If so, this may mean you have the option to *upgrade* your previously purchased mp3 file to a lossless format (i.e. pay a small fee, throw away the old mp3 file, and download the new lossless file).

Otherwise, what other posters have already said is true. You gain nothing by converting an mp3 file to a lossless format.