An iPod skipping songs doesn't have to mean your iPod is broken. Incorrectly synced or compressed files may not play on your iPod even if they play in iTunes, or the songs may not recognize your device as being authorized to play them. If all else fails, restoring your iPod to factory settings and re-syncing your library will fix many skipping issues.
If a song is synced to your iPod incorrectly, or if for any reason an individual audio file on your device has been corrupted, your iPod may skip over it completely. Re-syncing your iPod with your iTunes library may correct any damaged files. Connect your iPod to your computer, then wait for it to sync. If you need to make any adjustments to your sync settings, such as whether you want to sync only selected songs, do so after the sync is complete. Sync your iPod manually after you have saved your settings to make sure they take effect.
Re-import Specific Songs
If individual songs fail to play on your iPod, they may have been imported in the wrong format, even if they play correctly in iTunes. The songs may have also been corrupted or not compressed properly. Delete the songs from your iTunes library, then sync your iPod to ensure they have been removed from your device. Download them from your iTunes account or import them from your audio CDs and add them to your library and any playlists you want. Once the songs have been successfully re-added to your library, sync your iPod and try to play them again.
Deauthorize and Reauthorize Account
Any songs you buy using your iTunes account can only be played if you have authorized your computer with the same account. If a computer isn't authorized, the songs won't play. However, playback issues can sometimes occur even if your computer is authorized. To troubleshoot this, deauthorize your computer, then reauthorize it before syncing your iPod again. Click "Store," then click "Deauthorize This Computer." Click "Store" again after your computer is successfully deauthorized, then click "Authorize This Computer." You will need your email address and password associated with your iTunes account. Verify that your songs play correctly in iTunes, then sync your iPod again.
If your songs play without issue in iTunes but will not play on your iPod, your iPod may need to be restored. Restoring wipes all data on your iPod, then takes you through a setup process like it is a new iPod. Connect your iPod to your computer, select your iPod's name from the iTunes window, then click "Restore." Keep your iPod connected to your computer until the setup process is complete. After you name your iPod, it will sync all selected data based on your iTunes preferences, so it may take some time. Before disconnecting your iPod when the sync is complete, make sure the iPod's software is up-to-date.