Sounds like an old dryer or A/C outlet. You don't need to replace the 10 gauge wire unless it doesn't have a safety ground wire. But you will need to replace the circuit breaker with a single 20-amp breaker and a 20-amp receptacle. The circuit breaker will not permit more than 20 amps to flow before it trips which is the reason why 10 gauge wire is okay.
Before starting you need to ascertain if you have AL- aluminum wire or CU- copper wire. Hopefully it's all copper. If you have AL wire, you'll need to apply an anti-oxidant and devices marked AL/CU. Mixing aluminum and copper is a no no.
Now assuming you have a double breaker right now, you'll see a black wire & a red (10/3) or a white (10/2) wire on the double 220/240 breaker. It is hard to say without looking what you might find as some home owners do the darnedest things and end up electrocuting themselves or causing fires. Only you know your capabilities/experience and need to ensure you are within the code or should your place burn down you'll find yourself without a home and most likely without insurance coverage.
Be sure to turn the main circuit breaker off to the loadcenter/circuit breaker panel. Now even through you shut off the main circuit breaker, the large lugs where the main line is connected are still alive. Do not touch them. If you know how to use a voltmeter and have one, use it. You can also buy a device that will beep or chirp if any voltage is present like this:
http://www.amazon.com/Triplett-Non-Cont ... B0002THPMU
You also need to be sure you have a safe circuit breaker panel. There is one make which I can not recall which was discontinued because of inferior quality as I recall.
Now assuming you have all copper, unplug/remove the double breaker and unscrew the two wires. Screw the black wire into the new 20-amp breaker and the snap the breaker in or snap the new breaker in first & then attach the black wire. That leaves an extra wire- most likely a red (10/3) or white (10/2)with red tape or just a white (10/2). If it is fairly new, then it is possible to already have a 10/3 with a black, red, white, & a safety ground wire, but you have to look first. There is printing on the wire every foot or so that identifies the gauge and number of conductors. But even though it might say 10/3 and you see 4 wires, the safety ground is not counted as a conductor, so that is why it is called a 10/3.
If you have a 10/3 , just remove the red and cap it with a wire nut and tape the nut so it doesn't fall off & do the same at the receptacle. Who knows? Maybe you'll want to go back to 220/240 some day. At the receptacle, connect the black to the gold screw, then connect the white to the silver screw, and the bare copper wire to the green screw on the receptacle.
Using the 10 gauge wire will actually cause less of a voltage drop on the wire as everyone knows that electrons flow is on the outside of the conductor...<smile>...it actually does. The larger the conductor, the lower the wire resistance. People often wonder why devices which are connected to 2-3 extension cords as to why the extension cord feels warm or even hot, have dimmer lights, or don't sound like they do when plugged in directly to an outlet. The reason is most people buy cheap extension cords with 14 gauge wire. Had they used a more expensive contractor heavy-duty extension cord, then chances are the wire is at least 12 gauge or maybe even 10 gauge. The smaller the number, the thicker the wire and the less the voltage drop/resistance/the less heat created.
Here is a link:
http://www.doityourself.com/forum/elect ... gle.html#b
If you have a 10/2, then it will most likely be a black, white, and a safety ground (bare wire). Here is where you need to be more careful depending on if this is a main panel or a subpanel. In a main panel, the neutral wires and safety grounds are all connected together. However, on a subpanel, the neutrals and the grounds are isolated from each other to prevent something called a ground-loop. I'll let you Google that as it would take too long to explain. In this case, you use the black wire as above, but the white wire must be placed only on the neutral bar/bus. If you see the whites all on a separate bar, the grounds on another bar, and this is a subpanel, then that is how to do it, otherwise, you can connect it with the other white and ground ground wires.
I know a guy who actually has a 120/240 motor. He bought it from an estate sale I believe. That is the only one I have seen.
The breaker size is ultimately determines the circuit capacity. If you have questions, Home Depot or Lowes will usually have one person who will be able to draw you a diagram and get you what you need. If you could show me some pics, I would be better able to assist you as there is a lot involved even though it is only a few wires. I always error on the side of safety. If you have any questions/concerns, please ask & assume nothing.
You could always PM me and if you have a way to send pics, I will be better able to advise you. But ultimately, you are responsible for any work. I worked in electronics for 40 years and was only shocked once. It was during my first year. I learned very quickly.
I do not consider myself an electrician. Before I retired my last position was a radar engineer on those orange "spinny things" at major airports as my younger daughter used to call them. I did electrical work as needed as my primary position was to ensure air traffic controllers scopes were working so they could take naps...<smile>.