Multi-CPU was the first version: You’d have one or more mainboards with one or more CPU chips on them. The main problem here was that the CPUs would have to expose some of their internal data to the other CPU so they wouldn’t get in their way.
The next step was hyper-threading. One chip on the mainboard but it had some parts twice internally so it could execute two instructions at the same time.
The current development is multi-core. It’s basically the original idea (several complete CPUs) but in a single chip. The advantage: Chip designers can easily put the additional wires for the sync signals into the chip (instead of having to route them out on a pin, then over the crowded mainboard and up into a second chip).
Super computers today are multi-cpu, multi-core: They have lots of mainboards with usually 2-4 CPUs on them, each CPU is multi-core and each has its own RAM.
[EDIT] You got that pretty much right. Just a few minor points:
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Hyper-threading keeps track of two contexts at once in a single core, exposing more parallelism to the out-of-order CPU core. This keeps the execution units fed with work, even when one thread is stalled on a cache miss, branch mispredict, or waiting for results from high-latency instructions. It’s a way to get more total throughput without replicating much hardware, but if anything it slows down each thread individually. See this Q&A for more details, and an explanation of what was wrong with the previous wording of this paragraph.
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The main problem with multi-CPU is that code running on them will eventually access the RAM. There are N CPUs but only one bus to access the RAM. So you must have some hardware which makes sure that a) each CPU gets a fair amount of RAM access, b) that accesses to the same part of the RAM don’t cause problems and c) most importantly, that CPU 2 will be notified when CPU 1 writes to some memory address which CPU 2 has in its internal cache. If that doesn’t happen, CPU 2 will happily use the cached value, oblivious to the fact that it is outdated
Just imagine you have tasks in a list and you want to spread them to all available CPUs. So CPU 1 will fetch the first element from the list and update the pointers. CPU 2 will do the same. For efficiency reasons, both CPUs will not only copy the few bytes into the cache but a whole “cache line” (whatever that may be). The assumption is that, when you read byte X, you’ll soon read X+1, too.
Now both CPUs have a copy of the memory in their cache. CPU 1 will then fetch the next item from the list. Without cache sync, it won’t have noticed that CPU 2 has changed the list, too, and it will start to work on the same item as CPU 2.
This is what effectively makes multi-CPU so complicated. Side effects of this can lead to a performance which is worse than what you’d get if the whole code ran only on a single CPU. The solution was multi-core: You can easily add as many wires as you need to synchronize the caches; you could even copy data from one cache to another (updating parts of a cache line without having to flush and reload it), etc. Or the cache logic could make sure that all CPUs get the same cache line when they access the same part of real RAM, simply blocking CPU 2 for a few nanoseconds until CPU 1 has made its changes.
[EDIT2] The main reason why multi-core is simpler than multi-cpu is that on a mainboard, you simply can’t run all wires between the two chips which you’d need to make sync effective. Plus a signal only travels 30cm/ns tops (speed of light; in a wire, you usually have much less). And don’t forget that, on a multi-layer mainboard, signals start to influence each other (crosstalk). We like to think that 0 is 0V and 1 is 5V but in reality, “0” is something between -0.5V (overdrive when dropping a line from 1->0) and .5V and “1” is anything above 0.8V.
If you have everything inside of a single chip, signals run much faster and you can have as many as you like (well, almost :). Also, signal crosstalk is much easier to control.