system
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human
ELI5: How did Alan Turing's first conception of a "computer" work?

I am in the midst of reading Alan Turing: The Enigma, and I am loving it.  However, the biographer writes in the style that the reader has a strong grasp on unconventional mathematical concepts (Set theory, Group Theory, Computable Numbers, etc.).  I am knowledgeable in mathematics, nothing past calculus, but still, I cannot seem to be able to grasp what the book is trying to say about Turing's idea of a computer that 'reads' and 'writes' on blocked tape.
gpt
The way that a Turing machine works is that it has a long tape with information on it and a device that reads information off of the tape and can also write information onto it. The tape is divided into small blocks that each contain a piece of information. At some point in the tape there's a series of instructions, and the device reads the instructions one at a time and follows them.

The instructions contain commands about what to do with other information on the tape, and can sometimes tell the device to skip over instructions or jump back to previous instructions. There are many different sets of instructions that a theoretical device could understand, but Turing was interested in finding out which set of instructions was the absolute smallest while still making it possible to write instructions for basically anything.

The bare minimum set of instructions a device must understand to be able to follow basically any kind of procedure is called "Turing complete." Turing complete sets of instructions have commands for doing basic math on numbers on the tape, moving information from one part of the tape to another, comparing two things on the tape, and jumping from one instruction to another based off of that comparison.

Any Turing machine which has these instructions can, in theory, be given instructions on how to do literally anything. It can be given instructions on how to clean a room (obviously with the addition of some instructions that talk to other circuits the Turing machine is connected to), to come up with a theory for the most efficient way to clean a room, to simulate every atom inside of a messy room, or even simulate a human brain which will ponder the philosophy of cleaning rooms.

Turing complete instruction sets can contain more instructions than that, but they must have those kinds of instructions. An example of an architecture which is not Turing complete would be one that can only do math calculations but offers no way of skipping over instructions. If there is no way to skip over instructions then it can't make decisions because it always has to do every instruction and can't skip over the ones that are irrelevant.

In the modern world Turing machines are used universally for computing. Real world Turing machines do not have infinitely long tape like the theoretical version did, instead they have memory circuits. They store information as ones and zeros instead of just drawing it on the tape, and the device that deals with the instructions is the processor. Sometimes there will be more than one processor and they will cooperate to run multiple programs.