I am currently running Windows 8.1 and I was wondering if I am able to install Ubuntu to my external HDD (It already has stuff on it but it still has a lot of space) and swap between the 2 operating systems. If possible when I update Ubuntu and Windows 8.1 will the updates go to the HDD that the OS was installed 2. Also if I can boot ubuntu from a external HDD how do I swap between Windows 8.1 and Ubuntu?
If by "swap" you mean booting into the different operating system (OS), then yes, it is entirely possible. Just install Ubuntu on the external hard drive, then ask the bios to boot into the external hard drive by default. Upon installation, Ubuntu will install the Grub (bootloader) that will allow you to choose which operating system to start. If you unplug the external hard drive, it will boot straight into Windows.
However, if by swap, you mean actively flipping from one OS to the the other, well, you can't do that. It's not just two different programs running at the same time that you can flip between. The operating system (OS) is responsible for the interactions between the hardware pieces of the computer. The OS needs to have full control of the hardware at all times.
If you REALLY want to flip between the two systems, you can use a virtual machine. Basically, you can set, say, Ubuntu, to be able to start Windows inside itself. You would obtain the layers:
Hardware-> Ubuntu -> "faked hardware" -> Windows
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