Note: This answer covers the installation of IntelliJ IDEA. For an extended script, that covers more JetBrains IDEs, as well as help for font rendering issues, please see this link provided by brendan.
Furthermore, a manual Desktop Entry creation is optional, as newer versions of IntelliJ offer to create it on first startup.

I have my intellij int /opt folder. So what I do is:

• Download Intellij
• Extract intellij to /opt-folder: sudo tar -xvf <intellij.tar> -C /opt/ (the -C option extracts the tar to the folder /opt/)
• Create a Desktop Entry File called idea.desktop (see example file below) and store it anywhere you want (let's assume in your home directory)
• Move the idea.desktop from your home directory to /usr/share/applications: sudo mv ~/idea.desktop /usr/share/applications/

Now (in a lot) Ubuntu versions you can start the application after the GUI is restarted. If you don't know how to do that, you can restart your PC..

idea.desktop (this is for community edition version 14.1.2, you have to change the paths in Exec= and Icon= lines if the path is different for you):

[Desktop Entry]                                                                 
Encoding=UTF-8
Name=IntelliJ IDEA
Comment=IntelliJ IDEA
Exec=/opt/ideaIC-14.1.2/bin/idea.sh
Icon=/opt/ideaIC-14.1.2/bin/idea.png
Terminal=false
StartupNotify=true
Type=Application

Edit
I also found a shell script that does this for you, here. The given script in the link installs Oracle Java 7 for you and gives you the choice between Community and Ultimate Edition. It then automatically downloads the newest version for you, extracts it and creates a desktop entry.
I have modified the scripts to fulfill my needs. It does not install java 8 and it does not ask you for the version you want to install (but the version is kept in a variable to easily change that). You can also update Intellij with it. But then you have to (so far) manually remove the old folder! This is what i got:

Edit2
Here is the new version of the script. As mentioned in the comments, breandan has updated the script to be more stable (the jetbrains website changed its behavior). Thanks for the update, breandan.

#!/bin/sh

echo "Installing IntelliJ IDEA..."

# We need root to install
[ $(id -u) != "0" ] && exec sudo "$0" "$@"

# Attempt to install a JDK
# apt-get install openjdk-8-jdk
# add-apt-repository ppa:webupd8team/java && apt-get update && apt-get install oracle-java8-installer

# Prompt for edition
#while true; do
#    read -p "Enter 'U' for Ultimate or 'C' for Community: " ed 
#    case $ed in
#        [Uu]* ) ed=U; break;;
#        [Cc]* ) ed=C; break;;
#    esac
#done
ed=C

# Fetch the most recent version
VERSION=$(wget "https://www.jetbrains.com/intellij-repository/releases" -qO- | grep -P -o -m 1 "(?<=https://www.jetbrains.com/intellij-repository/releases/com/jetbrains/intellij/idea/BUILD/)[^/]+(?=/)")

# Prepend base URL for download
URL="https://download.jetbrains.com/idea/ideaI$ed-$VERSION.tar.gz"

echo $URL

# Truncate filename
FILE=$(basename ${URL})

# Set download directory
DEST=~/Downloads/$FILE

echo "Downloading idea-I$ed-$VERSION to $DEST..."

# Download binary
wget -cO ${DEST} ${URL} --read-timeout=5 --tries=0

echo "Download complete!"

# Set directory name
DIR="/opt/idea-I$ed-$VERSION"

echo "Installing to $DIR"

# Untar file
if mkdir ${DIR}; then
    tar -xzf ${DEST} -C ${DIR} --strip-components=1
fi

# Grab executable folder
BIN="$DIR/bin"

# Add permissions to install directory
chmod -R +rwx ${DIR}

# Set desktop shortcut path
DESK=/usr/share/applications/IDEA.desktop

# Add desktop shortcut
echo -e "[Desktop Entry]\nEncoding=UTF-8\nName=IntelliJ IDEA\nComment=IntelliJ IDEA\nExec=${BIN}/idea.sh\nIcon=${BIN}/idea.png\nTerminal=false\nStartupNotify=true\nType=Application" -e > ${DESK}

# Create symlink entry
ln -s ${BIN}/idea.sh /usr/local/bin/idea

echo "Done."  

Old Version

#!/bin/sh                                                                                                                                   

echo "Installing IntelliJ IDEA..."

# We need root to install
[ $(id -u) != "0" ] && exec sudo "$0" "$@"

# define version (ultimate. change to 'C' for Community)
ed='U'

# Fetch the most recent community edition URL
URL=$(wget "https://www.jetbrains.com/idea/download/download_thanks.jsp?edition=I${ed}&os=linux" -qO- | grep -o -m 1 "https://download.jetbrains.com/idea/.*gz")

echo "URL: ${URL}"
echo "basename(url): $(basename ${URL})"

# Truncate filename
FILE=$(basename ${URL})

echo "File: ${FILE}"

# Download binary
wget -cO /tmp/${FILE} ${URL} --read-timeout=5 --tries=0

# Set directory name
DIR="${FILE%\.tar\.gz}"

# Untar file
if mkdir /opt/${DIR}; then
    tar -xvzf /tmp/${FILE} -C /opt/${DIR} --strip-components=1
fi

# Grab executable folder
BIN="/opt/$DIR/bin"

# Add permissions to install directory
chmod 755 ${BIN}/idea.sh

# Set desktop shortcut path
DESK=/usr/share/applications/IDEA.desktop

# Add desktop shortcut                     
echo -e "[Desktop Entry]\nEncoding=UTF-8\nName=IntelliJ IDEA\nComment=IntelliJ IDEA\nExec=${BIN}/idea.sh\nIcon=${BIN}/idea.png\nTerminal=false\nStartupNotify=true\nType=Application" > ${DESK}

echo "Done."