A Y-adapter would at best give you mirrored displays (same thing on both, since it just splits the signal)
If you want both monitors to be different, then you would need something like the Matrox DualHead2Go, although it is rather expensive for what it offers.
If you want a cheaper route for multiple outputs, your best bet would be to get an add-in card (standard PCI would be the safe bet if you don't know what slots you have available inside the computer), since pretty much any normal video card will do dual outputs.
If you can't add a card, then the DualHead2Go is your only option if you want discrete displays, the Y adapter will only let you mirror.