My music will not download from iTunes onto my iPod.

Was the iPod previous synced to another iTunes library/computer?

Have you successfully synced from this iTunes library/computer before?

Do the songs play in iTunes?

Do you have the right boxes checked to sync?

iTunes: Syncing media content to iOS devices and iPod

Try syncing using the manual method

Managing content manually on iPhone, iPad, and iPod

- Restore from backup. See:

iOS: How to back up

That means that iTunes can't find the actual song files.

Try:

iTunes: Finding lost media and downloads

You can redownload most iTunes purchases by:

Downloading past purchases from the App Store, iBookstore, and iTunes Store

I though yo said the songs played in iTunes on the computer?

Did you restore the iPod?