I know vaguely that Norway and Sweden were united once, though I don't know when they were and when/why this ended. What caused Norway and Sweden to split and when, and why have they remained split?
As far as the other countries of Scandinavia, has there ever been any movement for a sort of "Scandinavian Union"?
The economic (and hence geopolitical) interests of the two countries were, and are, quite different; they are separated by a mountain range that makes it difficult to get much overland trade going. Sweden looks east across the Baltic (and south to Germany); Norway looks west, to the North Sea and the Atlantic (and to the UK). The union was relatively loose, consisting mainly of Norwegian foreign and trade policy being decided in Sweden; unfortunately the trade policy was precisely the place where many wealthy and influential men felt they could do a lot better, for the reason outlined above.
The split occurred in 1905. The immediate dispute was over what was called the "consulate question": The Swedish foreign ministry decided where to send ministers and what countries to prioritise, and was not answerable to the Norwegian Storting (parliament). Norway's merchant fleet was (and remains) quite large, and the ship-owners felt that their interests were not being looked after; there was lots of Norwegian trade outside Europe, but few consulates. So, the Norwegian government demanded the right to appoint its own ministers. The Swedes refused, though offering a compromise that in effect left them most of the power - basically they offered to give the Norwegians a formally "separate" foreign ministry, it would in effect have been subordinate to the Swedish one. This was referred to in Norway as the "vassal-state compromise" (loosely translated; more literally, the "satrapy points") and was rejected, leading to the breakup of the union. It may be worth noting that the consulate question had been a thing, on and off, for ten years, and that 1905 was, by some odd coincidence, a high point in Norwegian military power due to a llong-running armament program.
So, there are underlying economic interests, a direct expression of those underlying frictions in the dispute over what embassies and consulates should be prioritised, and then you have the more ideological point that the nineteenth century, and perhaps especially the turn of the century, was a period of very strong nationalism. The Norwegians felt (and feel) themselves a very separate people from the Swedes, although the difference is perhaps not so visible from the outside. :)
Has there ever been any movement for a sort of "Scandinavian Union"?
Sure. It's a perennially popular project among intellectuals; Ibsen was all in favour of it, for example. (And then there's the Kalmar Union, of course.) There's even one on reddit right now that's trying to make a Scandinavian alt-coin, called the eKrona, as the first step to forming a closer economic union.
Source: I learned this in school.
As far as the other countries of Scandinavia, has there ever been any movement for a sort of "Scandinavian Union"?
The Helsinki treaty was signed in 1962 to
... promote and strengthen the close ties existing between the Nordic peoples in matters of culture, and of legal and social philosophy, and to extend the scale of co-operation between the Nordic countries
So in a way, there has been "a sort of Scandinavian Union" for a while now.