If possible, you should replace the recepticle, like Roy said.
Hopefully the original installer ran something like 10/3 wire to it.
That means that it should have black, white, red, and bare (or green)
wires. Then, it's a simple switch-over to put in a four-prong
recepticle like a NEMA 6-20; just wire as the directions show. Then
connect green from your pigtail to the ground terminal on the
four-prong plug, and connect white and black where the directions say
to put red and black. The neutral terminal on the plug will have no
connection (even though the directions will probably say to connect the
white wire there).
However, given that there's only a 3-prong outlet now, there's a
reasonable chance that the installer only ran 10/2 (white, black, and
bare) to it. In that case, the RIGHT thing to do is to rerun 10/3 or
12/3 wire from the box, but that's a huge pain. It's very unlikely
that your lathe requires a neutral connection (and if it did, one would
hope that their supplied pigtail would have black, red, white, and
green). So if you're one to take shortcuts, keep your three-prong plug
and connect the green wire to the L-shaped prong, and the white and
black to the slanted prongs. The neutral should technically be
grounded back at your main panel.
If you really want to be sure, using a multimeter (sometimes called a
VOM), you should be able to measure a short circuit between the green
wire and any exposed metal on your lathe (thus verifying that the green
is, indeed, ground), and some relatively low resistance (maybe
somewhere between 0.5 and 5 ohms) between the white and black wires.
There should be an open circuit (very high resistance) between the
green and either of the other two.