It’s become easy to install Node.js on Raspberry Pi with Raspbian.

First of all, download the arm version of the latest Node.js here.
Change below to the version you downloaded.

#!/bin/bash

#Make a new dir where you'll put the binary
sudo mkdir /opt/node

#Get it
wget http://nodejs.org/dist/v0.10.4/node-v0.10.4-linux-arm-pi.tar.gz

#unpack
tar xvzf node-v0.10.4-linux-arm-pi.tar.gz

#Copy to the dir you made as the first step
sudo cp -r node-v0.10.2-linux-arm-pi/* /opt/node

#Add node to your path so you can call it with just "node"
cd ~
nano .bash_profile

#Add these lines to the file you opened
PATH=$PATH:/opt/node/bin
export PATH
#Save and exit

#Test
node -v
npm -v

EDIT
Thanks to johnmclear in the comments who pointed out a huge error in my code. I changed the instructions above so they are correct.