The first Turing machine
&quot;Turing machines&quot; (or &quot;a-machines&quot;) are a mathematical concept, not actual, physical devices. Turing came up with them in order to write mathematical proofs about computers, with the following logic:

Writing proofs about physical wires and switches is extremely difficult.
Writing proofs about Turing machines is (relatively) easy.
Anything physical wires and switches can do, you can build a Turing machine to do (*) (**).

But Turing never built an actual machine that wrote symbols on a paper tape. Other people have, but only as a demonstration: here's one you can make out of a business card, for example.
Why did he never build a physical Turing machine? To put it simply, it just wouldn't be that useful. The thing is, nobody's ever come up with a model of computation that's stronger than a Turing machine (in that it can compute things a Turing machine can't). And it's been proven that several other models of computation, such as the lambda calculus or the Python programming language, are &quot;Turing-complete&quot;: they can do everything a Turing machine can.
So for anything except a mathematical proof, it's generally much more useful to use one of these other models. Then you can use the Turing machines in your proofs without any loss of generality.
(*) Specifically, any calculation: a Turing machine can't turn on a lightbulb, for example, but lightbulbs aren't very interesting from a theory-of-computation standpoint.
(**) As has been pointed out in the comments, Turing's main definition of &quot;computer&quot; was a human following an algorithm. He conjectured that there's no computation a human can do that a Turing machine can't do—but nobody has been able to prove this, in part because defining exactly what a human mind can do is incredibly difficult. Look into the Church-Turing Thesis if you're interested.