• First, you must find the program you want to install in AUR page: aur.archlinux.org
• When you find it you must download the proper tarball file. To do so click the ‟Download tarball” link at the right side of the page.
  
• You must also read the PKGBUILD file by clicking the link ‟View PKGBUILD” because sometimes there are essential instructions for the installation. To tell the truth I don’t bother to read the PKGBUILD file. I read it only if the installation fails. I know that its not the proper way but still this is what I do….
• Uncompress the tarball. If you want to use the Terminal for this use the command
  tar -xvzf name.tar.gz
• Open the Terminal and go to the folder you created when you uncompress the tarball file.
• Compile the package using the command (without sudo):
  makepkg -s
• When the makepkg command will be completed install the program with the command:
  sudo pacman -U *xz

Be careful

If the program you want to install has dependencies that are not already installed you must installed them first as we describe above. If a program has many uninstalled dependencies its installation will take some time. This is the main reason why people uses tools like yaourt or packer to automate the installation of AUR programs. Those tools searches the AUR for the program we want, they download it, install the missing dependencies and then they install the program.

I described here how to install and use yaourt.