Processor

Virtual Processor - The Hardware entity to which the tasks are assigned, executing their programs.

Providing more cores to a partition just means more compute capacity, to handle multiple tasks and reduce any task's wait time.Still more compute task come from SMT Feature. 
Power 7 cores can concurrently executes upto 4 tasks per core.

Logical processor = virtual processor (vCPU), they are one in the same.  

Cores are hardware.  Cores are the number of processors within your physical CPU chipset.  (Multi-core processors).

Sockets are hardware.  Sockets are the number of processor sockets that your motherboard has. 

A motherboard with 2 sockets can support 2 processors.  A "dual-core" processor = 2 CPUs.  A quad-core processor = 4 CPUs.

you can never use more physical CPUs than virtual CPUs as defined in your LPAR. Even if you allocate one virtual processor to an LPAR and set it to be uncapped, you can’t run more than one physical processor because there would be no other virtual processors available.  So Physical CPU must be less then vCPU.

This way, you can limit the LPARs in your shared processor pools even if your LPAR is uncapped and there are 16 processors available in a shared processor pool. You still won’t be able to use more than one physical CPU because you only allocated one virtual CPU.

A virtual processor can represent from 0.1 to 1 of a physical processor. If you have one virtual processor, the range it can physically consume will never be more than one. If you have three virtual processors, you can use from 0.3 to 3, but never more than three.
It makes sense, as you’re basically giving your VM the illusion that it’s dealing with a physical processor. If it boots up, and sees three virtual processors, even if it’s running on 0.3 physical processors, it won’t see more than three processors.

Simultaneous multithreading (SMT). With POWER7 you can have four SMT threads, so the one virtual processor you set up will appear as four logical processors in your VM. If you were to turn off SMT, you would only see one logical processor.

Example - Dedicated Processor 

Total Managed System Processor - 16

Minimum - 4

Desired - 8

Maximum - 12

This is used in two case scenarios.

A) During Start of LPAR, it will look for desired number of processor that is 8. If it dont find, and find 5 or 6 cpu,then it will start, but if it finds less than 4, it would give warning msg and will not start.

More details can be find on IBM website.