Ben Eater has created an excellent 8-bit computer that is true to the essential architecture of modern computers, yet is simple enough to fit on a few breadboards. It uses DIP-switches and push buttons as inputs, and LEDs and 7-segment displays as (debug) outputs. Even step-wise execution by stepping the computer clock is possible, such that every part of the computer can be observed as it functions and the internal state and memory can be modified by switches.

He made it out of simple discrete components, and explains it all in a very complete and detailed way, going down to the transistor level, while also presenting the high level view. Each video handles one part of his computer architecture in a clear fashion. Other videos go into the electronic fundamentals onto which his higher level components are based.

Obviously this computer has no advanced features such as caches, pipelines or predictive execution, but it is similar in architecture to the computers that are really used in practice, unlike a Turing machine. (Though TMs made out of Lego are cool as well).

Even if you have a computer science degree, this will be a very educational video series.

His website is also highly recommended: Build an 8-bit computer from scratch.