Computers are basically built out of logic gates. Though this is an abstract idealization of the real physical machinery, it is close enough to the truth that we can believe it for now. At a very basic level, these things work just like true/false predicates. Or if you've ever played minecraft, it works a lot like redstone. The field which studies how to put together logic gates to make interesting complex circuits, like computers, is called computer architecture. It is traditionally viewed as a mixture of computer science and electrical engineering.
The most basic logic gates are things like AND, and OR which just take bits together and smash out some boolean operation between them. By creating feed back loops in logic gates you can store memory. One type of standard memory circuit is called a flip-flop, and it is basically a little loop of wire together with some AND gates and power to keep it stable. Putting together multiple latches lets you create bit vectors, and these things are called registers (which are what things like eax and ebx represent). There are also many other types of parts, like adders, multiplexors and so on which implement various pieces of boolean logic. Here is a directory of some circuits:
http://www.labri.fr/perso/strandh/Teaching/AMP/Common/Strandh-Tutorial/Dir.html
Your CPU is basically a bunch of these things stuck together, all built out of the same basic logic gates. The way that your computer knows how to keep on executing instructions is that there is a special piece of machinery called a clock which emits pulses at regular intervals. When your CPU's clock emits a pulse it sets off a sequence of reactions in these logic gates that causes the CPU to execute an instruction. For example, when it reads an instruction that says "mov eax, ebx", what ends up happening is that the state of one of these registers (ebx) gets copied over to the state of another (eax) just in time before the next pulse of comes out of the clock.
Of course this is a gross oversimplification, but as a high level picture it is essentially correct. The rest of the details take awhile to explain, and there are a few things here that I neglected due to unnecessary subtlety (for example, in a real CPU sometimes multiple instructions get executed in a single clock; and due to register paging sometimes eax isn't always the same thing; and sometimes due to reordering occasionally the way that instructions get executed gets moved around, and so on). However, it is definitely worth learning the whole story since it is actually quite amazing (or at least I like to think so!) You would be doing yourself a great favor to go out and read up on this stuff, and maybe try building a few circuits of your own (either using real hardware, a simulator, or even minecraft!)
Anyway, hope that answers a bit of your question about what mov eax, ebx does.
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