For teachers and students of macroeconomics
This is about how real exchange rates are determined in the medium term. So we abstract from the complications caused by sticky prices and monetary policy. However as anyone who understands uncovered interest parity knows, exchange rates in the short run depend crucially on expectations about medium term exchange rates, so the determination of medium term exchange rates is important whatever your time horizon.
The framework I use when teaching at masters level is the ‘new open economy’ (NOEM) approach, associated with Obstfeld and Rogoff in particular. A classic survey is by Philip Lane. If this framework could be summed up in one sentence, it would be this. In a world where most international trade takes place in goods sold in imperfectly competitive markets, the real exchange rate moves to equate the demand and supply for domestically produced output.  What follows is not about whether that framework is empirically useful, but why teaching it can avoid some confusions and pitfalls.
This concept was not of course invented by NOEM. John Williamson’s approach to determining equilibrium exchange rates, later taken up by the IMF and others, is based on the same idea. (See this earlier post for references. Williamson's work can in turn be seen as a development of the 'Swan diagram'.) Indeed I sometimes get annoyed that the NOEM literature typically ignores its antecedents. However one source for confusion is that the essentially empirical literature associated with Williamson focuses on the current account, rather than the supply and demand for domestic output. It does this because the current account is a readily available indicator of this supply and demand balance much of the time. But not always, as the following classic example shows.
Suppose an economy discovers a finite natural resource, like oil, which takes a negligible amount of labour to extract. It takes a few years before the discovery leads to the resource being extracted, but the extent of the resource is common knowledge. This is a standard exercise in consumption smoothing. Consumption rises the moment the resource is discovered, anticipating higher future income. This leads to a current account deficit until the resource is extracted. Once it starts being extracted, consumers are now consuming less than their income, first to pay off their borrowing, and then to save for the day the resource runs out. So while the resource is extracted we get a current account surplus.
What happens to the real exchange rate? If we focus on the current account, we might be tempted to say that it first depreciates, and then appreciates when we have a surplus. This would be wrong. We could start with a special and highly unrealistic case, where there are no non-traded goods, the economy is so small that only a negligible amount of the additional consumption is spent on home produced goods, and labour supply is fixed. In that case nothing would happen to the real exchange rate at any time. More realistically, transport costs will mean there is some ‘home bias’ in consumption, and also some of the consumption will go on domestically produced non-traded goods. Both imply a domestic real appreciation, which begins while the current account is in deficit, and which stays the same as the current account switches to surplus. In addition, if consumers want to match higher consumption with more leisure, labour supply will decrease, and we get an appreciation to choke off demand for domestically produced goods. Again this happens throughout, and not just when the resource is extracted.
The reason why looking at the current account is misleading is that we are ignoring the capital account. Before the resource is extracted, consumption rises through borrowing from abroad. If all the extra consumption is on overseas goods, those lending to consumers require no domestic currency (they can lend in dollars). But if some of the additional consumption is spent domestically, some of the lending must also be in domestic currency, so we get an appreciation. Once the resource begins to be sold (for dollars), it is as if all the extra income is used to buy overseas assets. So the size of the appreciation remains unchanged.
Thinking about both current and capital accounts in this situation is tricky, but thinking about the supply and demand for the domestically produced tradable goods gives us the same answer much more easily.
 In a simple model without capital, supply is just labour supply and productivity. For a small open economy where there are no non-traded goods or home bias, demand for domestically produced goods just depends on world output and competitiveness=real exchange rate. In this simple set-up a consumer price based real exchange rate is constant (PPP holds), but once we introduce realistic features like home bias or non-traded goods competitiveness influences a consumer price based real exchange rate, and PPP no longer holds.
 For simplicity ignore the capital required to extract the resource, and we assume all the income from the resource goes to domestic consumers.
 The two mechanisms work in different ways, however. The additional demand for non-traded goods takes labour away from traded goods production, so reduced traded goods supply leads to an appreciation. With home bias we get an appreciation because of the additional demand for domestically produced traded goods.