You can burn a CD-R with whatever you like but only an Audio CD will play in your player.
Your music source can be mp3 or wav and most apps will automatically convert to Audio CD format (wav) for you.
I don't use Nero for this at all , my preference is Roxio Media Creator but there is a very good , free alternative called Burrrn (http://www.burrrn.net/) which is very simple to use.
Just add you music files & burn it.
- Which format of music should I download from allofmp3.com? (mp3, WAV etc.?)
Get them as mp3 as it's so much quicker.
- What's the difference between a CD-R audio disc and an MP3 disc? Meaning to burn a CD-R disc, what type of files or format of files do I need? If i want to make an MP3 disc, i can only do so with MP3 files?
Quite a lot, unless your player specially mentions that it plays mp3 , and many do these days, then it won't. For an ordinary player you need an Audio CD but you can have a source that is mp3 & it'll get converted for as part of the burn process.