Q: what does manually manage music and videos mean

Only music and other media that you drag over from your iTunes Library to your iPad will be synced to your iPad.

Your other options are to sync everything, and to sync everything except for items which don't have a check mark ("sync only checked items").

It means that you choose the music & video in iTunes to sync manually opposite of automatically.

It also means you haven't found the user guide, available at http://support.apple.com/manuals/ or downloadable from iTunes as an iBook.

Ignore. (Glitch - double posting)

I didn't find this very helpful.  While the manual technically states what it means, these types of questions are typically looking for context, helpful hints about WHY you would use it and potential issues you might have if you used it or didn't use it.  Your snarkiness doesn't help either.  Some of us are looking for actual help with the entire feature.  So I've pasted what I found in the manual link you posted showed (for iPad 8.1) as well as a better answer I found on Yahoo.

You can also choose to manually manage content from iTunes by selecting that option in the iPad Summary pane. Then you can drag songs or videos from your iTunes library to iPad in iTunes. This is useful if your iTunes library contains more items than can fit on your iPad. Note: If you use iTunes Match, you can manually manage only video.

https://answers.yahoo.com/question/index?qid=20080210005100AA4pJ6G

It is the best and easiest way to use your ipod. When you manually manage your music you choose what to add and delete from the ipod. You dont have to sync your ipod. When it is on manual you can use your ipod on as many computers and itunes libraries as you want without losing any music off your ipod. You just drag and drop the songs, videos and playlists you want from the itunes library to the ipods name. Right click and delete what you dont want from the main playlists under the ipods name.

Nothing will ever be deleted or added until you direct it and you can safely use it on any computer. If you click manually manage music nothing will change on your ipod as it is now until you physically add or delete things. If you resync with an itunes library later, though, you have to be careful because anything not in that itunes library will be deleted off the ipod. I always keep my ipod manually managed because I use it on several diff computers.

or you just highlight the 20 songs together and drag them in one action...some people are so lazy, lol.  Think if it this way If it is manually managed and you go to your friends house you can use her itunes library which means basically getting music from any other itunes library you want. Not just your own.

Thanks for that. NOW I understand what 'what does manually manage music and videos mean. ' I found the manual confusing but this explanation helped.