Currency carry trade refers to the trade, where an investor borrows a currency from a country with a low interest rate, and then invests it in a country that yields a higher interest rate.
The Yen carry trade is the most popular example of the currency carry trade, because Japan has had low interest rates for a fairly long period now. Investors borrowed cheap Japanese Yen, and then bought securities like the US Treasury Bills, which gave them relatively higher interest rates. This type of thing worked really well when exchange rates were stable, and traders could leverage their investments to get great returns. But, the downside is that since two currencies are involved, fluctuations in exchange rates can kill all and any profits you make on your trade.