What's the difference between a republic and a federation?

In short, there isn't one. If you look at forms of government as a sort of pyramid, at the top you'd have democracy and fascism etc (the big forms of governance), then in the middle you'd have republic, monarchy etc (what type of governance) and then at the bottom you'd have federation and other ways of running this country (the practicalities). A republic is fundamentally just a democratic country that is run by the people (as opposed to a monarchy, where the ruler gets their authority from God). A federation can be both a republic and a monarchy, it's just the logistics of running the country. Essentially a federation is the spreading of power between the central national government and regional governments, like in the USA. In the UK we're not quite a federation, because the Welsh, Northern Irish and Scottish government's aren't all equally autonomous and the UK parliament still has the biggest voice by far and large.