I have downloaded Ubuntu 2 times for VirtualBox but in VirtualBox it just shows the 32-Bit options in the operating system menu. This is why I am assuming that the Ubuntu ISO that I downloaded was a 32-Bit version. I get a message this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot” error. That's why I want to try the 64 bit version and I do have a 64 Bit system.
You can download all (including the 64-bit) editions of Ubuntu from : releases.ubuntu.com
Here is the direct download link for : Ubuntu 16.04.2 desktop 64-bit edition (5 years support)
Here is the direct download link for : Ubuntu 16.10 desktop 64-bit edition (9 months support)
Important : Make sure that the host machine on which you want to run the virtual guest system is running a 64-bit operating system that supports virtualization technology and that virtualization support is activated in the BIOS of this machine when you are using the VirtualBox application.
Updated information : The direct download links are valid only until a new (point) release will be available - as of today (2017-02-20) these are currently Ubuntu 16.04.2 LTS and Ubuntu 16.10. If one wants to download older (point) releases, they are located here : old-releases.ubuntu.com
You can get a 64 bit version on http://releases.ubuntu.com/16.04.1/
Unless your CPU supports virtualization (Intel VT-x or AMD-v), and that is enabled in BIOS, you can't install 64bit guest OS's in Virtualbox.
To check if the cpu supports virtualization, run
cat /proc/cpuinfo| egrep "vmx|svm" if it shows vmx or svm it is supported, and you only have to enable it in bios.