How do we define what makes a currency?
Money is generally defined as
Medium of exchange
Store of value
Unit of account
Anything that can be used as money is money. As such, money is flexible. If you think tulips, receipts (bank drafts, post-dated checks, warehouse receipts, tobacco , tulips, Dot Com stocks, cows, garbage pail kids, are money , it is. If you change your mind it does away. Currency tends to be "harder" – harder to create and destroy and tend to be tightly controled by a central bank. Read up on Money Supply and the difference between M0, M1, M2, etc. The further you go out the less something is like currency.
Does a currency need to be backed by some kind of country?
Technically no. Private currency has a long and rich history. However, since 1880 it has be on the decline. I can't think of any major currencies that are not backed by a government today.
Is there an expectation of stability of price?
Yes, but you don't always get what you want. Money (and thus currency) is supposed to be a low risk / low reward asset. Currency has low risk (see inflation) and has either 0% reward or more commonly a negative rate of return. (if not from inflation, then convince charges to hold and transfer funds. A advantage of BitCoin is that they will reduce the charges.) See countries with high inflation rates. Venezuela is my current favorite example. Argentina comes in a close second. FYI, the human brain from an organic aspect can handle inflation better than deflation.
Do you need an area of economic activity where the currency is ubiquitously accepted as a form of payment?
Mostly yes. It must be a unit exchange, barter, ability to pay taxes and loans, etc. If not it tends to gimp the money, leading to inefficiencies, eroding its value. For examples, you can look at some communist countries that have issued dual currency, one that is convertible into foreign currencies and one that is not. Food stamps that can only be used to buy food and the resulting black market in baby formula.