What is the difference between Full, Para and Hardware assisted virtualization?

Paravirtualization is virtualization in which the guest operating system (the one being virtualized) is aware that it is a guest and accordingly has drivers that, instead of issuing hardware commands, simply issue commands directly to the host operating system. This also includes memory and thread management as well, which usually require unavailable privileged instructions in the processor.

Full Virtualization is virtualization in which the guest operating system is unaware that it is in a virtualized environment, and therefore hardware is virtualized by the host operating system so that the guest can issue commands to what it thinks is actual hardware, but really are just simulated hardware devices created by the host.

Hardware Assisted Virtualization is a type of Full Virtualization where the microprocessor architecture has special instructions to aid the virtualization of hardware. These instructions might allow a virtual context to be setup so that the guest can execute privileged instructions directly on the processor without affecting the host. Such a feature set is often called a Hypervisor. If said instructions do not exist, Full Virtualization is still possible, however it must be done via software techniques such as Dynamic Recompilation where the host recompiles on the fly privileged instructions in the guest to be able to run in a non-privileged way on the host.

There is also a combination of Para Virtualization and Full Virtualization called Hybrid Virtualization where parts of the guest operating system use paravirtualization for certain hardware drivers, and the host uses full virtualization for other features. This often produces superior performance on the guest without the need for the guest to be completely paravirtualized. An example of this: The guest uses full virtualization for privileged instructions in the kernel but paravirtualization for IO requests using a special driver in the guest. This way the guest operating system does not need to be fully paravirtualized, since this is sometimes not available, but can still enjoy some paravirtualized features by implementing special drivers for the guest.