Gotham is the codename for XBMC Media Center version 13. Current release date is unknown and will be communicated through official channels when decided. You will often see "v13" and "Gotham" used interchangeably in the XBMC community.
For most users, no. All settings and your library will be preserved in an upgrade. Settings that aren't preserved are those that have been reorganised, renamed, set to a new default - usually it's only a handful, and only things that don't really matter. However, it never hurts to do a backup of your userdata folder (where all the settings/library DB files are stored).
Gotham now has multiple user levels for settings. Hiding some more advanced settings by default. To change this setting level just select the button/label that says "Settings level". Not all skins have been adapted to this change. If you cannot find this side panel, you must switch to the Confluence skin to enable these options. After this you can switch back to the skin you normally use.
The LCD/VFD functionality in XBMC has been spun-out into an add-on. Full functionality for LCD/VFD displays is included, plus new improvements and bugfixes. See Add-on:XBMC LCDproc for more details.
Linux
How to switch from XBMC unstable PPA (used in XBMCbuntu Beta/RC releases) to final PPA
Add-ons may not work, prompting XBMC to ask you to disable them. You can keep using them if you wish (by choosing not to disable them), but doing so is unsupported. See Category:All add-ons for a list (check the drop-down menus). Where there is a noticeable slowdown in XBMC addons are likely cause, backup your userdata and as a last resort delete the userdata Addons folder and the Addons15.db in the userdata Database folder.
Skins need to be updated for Gotham - check your skin has been upgraded ready for Gotham: Category:Skins (check the drop-down menus). If you have problems starting up XBMC after updating, you might need to move/delete your guisettings.xml file (see userdata for location) to reset your skin.
If videos are playing back slowly or you have having sound related issues, see: #Audio.
What happens if I don't like XBMC v13, can I just install XBMC v12 again?
Whilst Gotham does "upgrade" the library, it stores a Frodo compatible copy. Just replace the Gotham application with Frodo. It is still recommended to always make a backup of your userdata folder before updating, just in case. Add-ons will likely need to be reset (uninstall and then reinstall) if you go from Gotham to Frodo.