XAMPP is currently only available as 32 bit application. Please use a 32 bit compatibility library for your system.

Solution - I
If you have latest 64 bit Linux Operating system like recent versions of Ubuntu / Debian / CentOS or Fedora and you want to use older version of php or mysql ( php 5.3 or php 5.4) then you should follow Solution - I.

Older libraries of xampp supports 32 bit operating system so your latest 64 bit operating system is not able to install older xampp.

To make it working install Ubuntu 12.04 LTS or Debian 7.0 in Vmware or Virtualbox
or 
Download ubuntu 12.04 docker image and install xampp inside it, it is faster solution and will not occupy too much RAM.

Solution - II

If your xampp was working fine and suddenly it has started giving this error then this solution is for you

Most common reason of this error, you have added an extension in xampp and after adding it in php.ini, the lampp is not getting started. You are getting following error :

XAMPP is currently only available as 32 bit application. Please use a 32 bit compatibility library for your system.

Here is the solution for you.

Comment following code in the file /opt/lampp/lampp

#case `uname -m` in
#    *_64)
#    if $XAMPP_ROOT/bin/php -v > /dev/null 2>&1
#    then
#        :
#    else
#        $GETTEXT -s "XAMPP is currently only availably as 32 bit application. Please use a 32 bit compatibility library for your system."
#        exit 1
#    fi
#    ;;
#esac

Now restart xampp and you should not get above error.

If still there is error and it was because of new installed extension, I would suggest to comment the extension .so line in php.ini and restart xampp.

You can see the solution in the following video.