How can a country manipulate its currency?
For example: Canada actively purchases US dollars in an effort to control the US/Canadian rate.
What are the ways Canada can use purchased USD to benefit the Canadian dollar?
By buying USD, the Canadian government reduces the number of USD in the money supply, thus making the USD rarer and so more valuable compared to the CAD.
This dollar buying also increases the number of USD in its own central bank, and that increases the "national value" of Canada, which increases the value of its own currency; however it can mitigate this, if it wishes to have a cheaper currency than the USD, by simply printing more.
To answer the broader question, there are many ways a country can manipulate its currency:
This question is built on a faulty premise. It's impossible to manipulate a single exchange rate like USD1 to CAD2 without affecting other exchange rates.
This is because of arbitrage; the fact that both currencies can be exchanged into a third currency. Here's a simple example explaining why:
Let's say that the USD is trading at parity to the CAD; they both have the same value. And let's say that a USD (or CAD) is worth 0.7 EUR3.
Assume that the United States government starts buying CAD and selling USD. Let's assume that they buy enough CAD so that one CAD buys 1.05 USD.
That has to affect all exchange rates, otherwise you could transfer your USD into EUR, and then into CAD and magically4 make money. Arbitrage5 prevents this from happening.
In general Currency Manipulation occurs through:
1) Raising interest rates to encourage people to buy your currency so that they can earn a high rate of return.
2) Intervention into the foreign current markets buying/selling a currency.
3) One of the purest forms of manipulation is threatening to intervene. Here's an example in Japan.
1. United States Dollar
2. Canadian Dollar
4. One USD is worth 0.7 EUROS which is worth one CAD which is worth 1.05 USD.
5. In this example arbitrage would be exchanging USD -> EUR -> CAD -> USD and making 5% on every dollar.