I want to remove Videos and Music from the left side panel. But, I can't seem to find the option to do so. How do you do that?

Tested in Ubuntu 14.04
Those bookmarks are controlled by ~/.config/user-dirs.dirs and commenting out a lines configures the appearance of the list
As an example, to remove the Videos bookmark from the list change the line:
XDG_VIDEOS_DIR="$HOME/Videos"
to
#XDG_VIDEOS_DIR="$HOME/Videos"
Note:
After login the file ~/.config/user-dirs.dirs gets reverted to the original set by /etc/xdg/user-dirs.defaults. To prevent that make the file "read-only" either by right clicking and then properties > permissions or using the command
chmod -w ~/.config/user-dirs.dirs
If you need to edit the file again do the opposite via Nautilus or use the command
chmod +w ~/.config/user-dirs.dirs
If you want to apply this to all users in your system change the file vim /etc/xdg/user-dirs.conf and set the option enabled to False. You need root permissions or this.
Open the file user-dirs.dirs in your ~/.config folder with your favorite text editor.
Comment out the line about the folder, which you do not want to be in the nautilus left pane. I commented about the Videos folder.
.......... .......... ......... XDG_DOCUMENTS_DIR="$HOME/Documents" XDG_MUSIC_DIR="$HOME/Music" XDG_PICTURES_DIR="$HOME/Pictures" #XDG_VIDEOS_DIR="$HOME/Videos"
Then open a terminal, and run xdg-user-dirs-gtk-update, see the magic.
In Nautilus >= 3.6 this will not survive a logout/login or reboot. To overcome this we have to point our XDG directories to "$HOME" like e.g.:
XDG_VIDEOS_DIR="$HOME"
Aha - the definitive ~answer~ work-around, thanks to A. J. McMinn: Removing entries from Nautilus Places
Comment out the unwanted bookmarks in ~/.config/user-dirs.dirs
Make or edit a ~/.config/user-dirs.conf file and add the entry enabled=false.
This solution persists over boot.
*...this could be done with a one-liner: echo "enabled=false" > ~/.config/user-dirs.conf
These directories are set by xdg-user-dirs.
Reading the documentation shows that you can disable a user directory by pointing it to your home directory. Use Ubuntu-Tweak or manually edit the file ~/.conf/user-dirs.dirs and point all the bookmarks you do not want to see to your home dir, and they will also disappear from the Nautilus side bar.
NOTE: I tried commenting out and removing the lines from ~/.conf/user-dirs.dirs as forestpiskie suggests, and that worked only until the next time I logged in. By setting them to the $HOME dir, the settings stuck.
The Files (Nautilus) application reads a list of directories from the file ~/.config/user-dirs.dirs in your home directory. These directories get added to the side-bar, and cannot otherwise be removed.
Here are the default contents of ~/.config/user-dirs.dirs if your locale is in English:
# This file is written by xdg-user-dirs-update
# If you want to change or add directories, just edit the line you're
# interested in. All local changes will be retained on the next run.
# Format is XDG_xxx_DIR="$HOME/yyy", where yyy is a shell-escaped
# homedir-relative path, or XDG_xxx_DIR="/yyy", where /yyy is an
# absolute path. No other format is supported.
#
XDG_DESKTOP_DIR="$HOME/Desktop"
XDG_DOWNLOAD_DIR="$HOME/Downloads"
XDG_TEMPLATES_DIR="$HOME/Templates"
XDG_PUBLICSHARE_DIR="$HOME/Public"
XDG_DOCUMENTS_DIR="$HOME/Documents"
XDG_MUSIC_DIR="$HOME/Music"
XDG_PICTURES_DIR="$HOME/Pictures"
XDG_VIDEOS_DIR="$HOME/Videos"
You may be tempted to delete the lines that you don't want, but this won't work. When you log in to your GNOME session, the command xdg-user-dirs-update is run, and it will add back any lines that you've deleted. The main purpose of this command (along with xdg-user-dirs-gtk-update) is to check that these directories exist, and that their names are the expected names for the current locale.
If you want to want to make sure that ~/.config/user-dirs.dirs remains unchanged, you need to change a different file named ~/.config/user-dirs.conf (notice the filename extension is .conf, not .dirs this time), and add enabled=false (with no spaces), with this command:
echo 'enabled=false' > ~/.config/user-dirs.conf
Now comment out or delete any lines you want from ~/.config/user-dirs.dirs . Run xdg-user-dirs-update to verify that it remains unchanged. Run nautilus --quit to close any existing Files processes, and then open the Files application.
(Alternatively, you can modify the default directories that xdg-user-dirs-update fill in by modifying /etc/xdg/user-dirs.default (this requires sudo privileges and it affects all users on that computer). After doing that, modify ~/.config/user-dirs.dirs, run xdg-user-dirs-update, and restart Nautilus.)
You may notice that these folders are now listed in the third section of the side-bar, but this time, you can easily remove them, by right-clicking on them, and clicking "Remove from Bookmarks". You can finally get this result seen in this screenshot:
Note that Files (Nautilus) is not the only application that uses ~/.config/user-dirs.dirs , so if any other application expects to find a folder for Documents or Pictures or Videos or Music, it may have unexpected behaviour.
There are rumours that GNOME developers are considering making it easier for users to remove default directories from the side-bar, according to this OMGUbuntu article in 2023. In 2024, some code got committed to make this change.
~/.config/user-dirs.dirs like
comment out the directories you don not want for example.echo "enabled=false" > ~/.config/user-dirs.conf which will create the user-dirs.conf file containing the option enabled=false.
These steps still seem to work in 18.04. Based on comments in this thread I made the following bash script that performs the task. Note, you still need to restart nautilus and right-click remove the items after executing these commands:
nautilus_hide_unwanted_sidebar_items()
{
echo "Removing unwanted nautilus sidebar items"
if [ "1" == "0" ]; then
# Sidebar items are governed by files in $HOME and /etc
ls ~/.config/user-dirs*
ls /etc/xdg/user-dirs*
cat ~/.config/user-dirs.dirs
cat ~/.config/user-dirs.locale
cat /etc/xdg/user-dirs.conf
cat /etc/xdg/user-dirs.defaults
#cat ~/.config/user-dirs.conf
fi
### --------------------------------------
### modify local config files in $HOME/.config
### --------------------------------------
chmod u+w ~/.config/user-dirs.dirs
#sed -i 's/XDG_DOCUMENTS_DIR/#XDG_DOCUMENTS_DIR/' ~/.config/user-dirs.dirs
sed -i 's/XDG_TEMPLATES_DIR/#XDG_TEMPLATES_DIR/' ~/.config/user-dirs.dirs
sed -i 's/XDG_PUBLICSHARE_DIR/#XDG_PUBLICSHARE_DIR/' ~/.config/user-dirs.dirs
sed -i 's/XDG_MUSIC_DIR/#XDG_MUSIC_DIR/' ~/.config/user-dirs.dirs
sed -i 's/XDG_PICTURES_DIR/#XDG_PICTURES_DIR/' ~/.config/user-dirs.dirs
sed -i 's/XDG_VIDEOS_DIR/#XDG_VIDEOS_DIR/' ~/.config/user-dirs.dirs
###
echo "enabled=true" >> ~/.config/user-dirs.conf
chmod u-w ~/.config/user-dirs.dirs
### --------------------------------------
### Modify global config files in /etc/xdg
### --------------------------------------
#sudo sed -i 's/DOCUMENTS/#DOCUMENTS/' /etc/xdg/user-dirs.defaults
sudo sed -i 's/TEMPLATES/#TEMPLATES/' /etc/xdg/user-dirs.defaults
sudo sed -i 's/PUBLICSHARE/#PUBLICSHARE/' /etc/xdg/user-dirs.defaults
sudo sed -i 's/MUSIC/#MUSIC/' /etc/xdg/user-dirs.defaults
sudo sed -i 's/PICTURES/#PICTURES/' /etc/xdg/user-dirs.defaults
sudo sed -i 's/VIDEOS/#VIDEOS/' /etc/xdg/user-dirs.defaults
###
sudo sed -i "s/enabled=true/enabled=false/" /etc/xdg/user-dirs.conf
sudo echo "enabled=false" >> /etc/xdg/user-dirs.conf
sudo sed -i "s/enabled=true/enabled=false/" /etc/xdg/user-dirs.conf
# Trigger an update
xdg-user-dirs-gtk-update
echo "
NOTE:
After restarting nautilus the unwanted items will be demoted to regular
bookmarks. You can now removed them via the right click context menu.
"
}
Edit -
Dug into it a bit further - if you edit ~/.config/user-dirs.dir you can remove them from the panel and still have them as folders in your /home it seems.

try going on Bookmarks in window menu, bookmarks > modify bookmarks (translate from italian ubuntu) and the remove the one that you don't want
otherwise Ctrl + D to access the menu