A voucher is kind of like a gift certificate. You would use it as partial (or complete) payment of a future airfare. On AA, vouchers are paper, so you would take them to your airport or city ticket office (or maybe your travel agent). Southwest (and maybe others) use e-vouchers--the money is recorded under a six-letter code.

There might be a problem in using a voucher from before you come home here--the voucher would be in ¥, but the ticket here would be in US$. I would think they would take it at a reasonably current exchange rate, but I'm not entirely certain on that point.