Steam Client

To my knowledge the information stays on your account, but to be safe you can back up this file:
STEAM/userdata/$UID/7/remote/sharedconfig.vdf

Or just copy the entire Steam folder.

The categories for your Steam games are stored in Steam/userdata/[userID]/7/remote/sharedconfig.vdf.
The categories for your non-Steam games are stored along with the shortcuts themselves, in userdata/[userID]/config/shortcuts.vdf.
The Steam categories ARE synced, but not on a continuous basis. As far as I know, they are ONLY saved to the server when Steam closes, and are ONLY downloaded from the server when it starts. Things may get wonky when you have Steam running in multiple locations. If you're going to be making changes through the client, I find that it's best to make sure that Steam is only running in one place.
If in doubt, it's that 7/remote/sharedconfig.vdf that you want to back up, NOT config/localconfig.vdf.

From:
https://www.reddit.com/r/Steam/comments/2c8874/sync_of_libraries_categories_between_pcs/

You'll keep all the categories and stuff. They're in Steam Cloud.

Hey hey if you would like to keep everything exactly as it is (your installed games and such included) copy over your whole steam folder onto the new PC and then install steam off of the downloaded install file and install it into the copied steam folder.

Thanks everyone, this is great news, i was not looking forward to having to re-organize my library, have a great Christmas to those that celebrate it.