Looking for some more details on "Turing" devices I've been reading a lot of computer science literature in the recent past but haven't ran across an explanation of Turing machines, the different types, and why they seem to come up so often (I understand they are a sort of a basis for CS topics).  

I've also seen comments by some members mentioning turing-completeness (Particularly when dealing with programming languages) and was just looking for some clarification.  

I did search around before posting this and haven't been able to find a similar post or a website that describes this in terms that I have been able to understand or decipher.  Thanks.
A Turing machine is a simple theoretical/conceptual computational device based on state.  Computation is done through state transitions based on the machines current state.  It is popular in Computer Science theory due to how models and proofs can be built off of it.  Wikipedia has a far more detailed write up on Turing machines than is practical to put here and a general discussion of Turing machines is probably off topic for IT Security.  Computer Science is probably a good place to ask more about Turing machines in general, since they are a fundamental conceptual device for Computer Science theory.

In case you are a visual-learner, interestingly enough, there exist physical incarnations of the abstract device.