A herring is a kind of fish. Apparently it has V-shaped bones, because V-shaped is what "herringbone" means in clothing design. So I'm guessing that a "herringbone machine" is one in which there are lots of opportunities for the flow of control to go in two different directions.
I like hydraulic computers, in which water under pressure (not a lot of pressure; don't think of firehoses) is used to drive mechanical action.
Some people have, apparently seriously, proposed mechanical computers for use after a nuclear attack, which would destroy all electronic devices in the vicinity. (Somewhat relatedly, after every large-area power failure, there are news stories about people using an abacus to compute payments in their stores.)
I once, long ago, got to play with an analog computer, which is electronic, not mechanical, but still very different from the computers we all know. It uses the voltage on a wire to represent a value, so 1 volt means 1, 2 volts means 2, 1½ volts means 1½, etc. You only get two or three digits of precision, but you also get quick and easy solutions to differential equations, so in the early days of digital computers people still preferred analog computers for certain applications. (Now you don't have to write the software for math problems any more, so everyone prefers the much better precision of digital computers.)
To @bh: I saw a picture of a herringbone machine, and it looks like it's made of sticks very much like popsicle sticks that are stood upon supports and themselves. I'd have to find the topic that has a link to the website I found it, though.Here's the website. You'll have to scroll down a little bit to see the herringbone OR gate.
To @sir_kitten2: I'm not sure. It does have a few ~30 second videos, but if, say, Youtube works fine, then that likely isn't it.
You can make a water powered computer, something Steve Mould made a while back. You can also try to design your own gear-powered computer, lever-powered computer, or an electromechanical computer. (If you are successful I would LIKE to see you divide 0 with 0 on it and see if it BLOWS up!)
I am on the verge of making a fully mechanical AI as I believe AI is all about MEMORY and LEARNING FROM PAST MISTAKES and not about all that neural network stuff or machine learning. Also this has never been done before so I'll try to make one. I won't call it WORLD'S FIRST as that would be too naive of me. Also I like mechanical machines so project is a go.
"Machine learning" is a general term for any program whose algorithm is refined by experience. My Animal Game project is a simple example.
"Neural networks" is the name of a particular learning system that tries to model what we know about how the human brain works, with neurons that store state, synapses that connect the output of one neuron to an input of another, and a weighting function in each neuron that controls how much each input synapse contributes to setting the state of that neuron. This has been a very successful learning model in a wide range of applications.
But in practice the two terms are widely used interchangeably, because nobody does machine learning any way other than neural nets these days. That's not to say that @slate_technologies shouldn't build simpler machine learning systems; it's just a thing to know about what people are doing out there in the Real World.
Getting a neural net to work involves some pretty hairy math to determine the best weighting functions. (And that's about as far as my knowledge goes.)
When I was a kid there were electric adding machines with wheels to enter numbers and read results. When you multiplied, it did the same shifted-partial-products algorithm you learned in elementary school.
And when you told it to divide by zero, it'd go into an infinite loop and would have to be reset. And the secretaries in the office would get mad at you. :~)