Will my American Xbox work in China?

As I understand it there are various categories of device that people use when taking equipment abroad.

• Plug adaptors simply convert the type of the plug, they do not change voltage. So they can only be used if the device is compatible with the voltage of the destination country.
• Thyrister or Triac based converters, work in a similar way to a lighting dimmer. They reduce the RMS voltage but they do not proportionally reduce the peak voltage (and possibly do not reduce it at all). This is ok for heating elements but use with other devices may cause damage.
• Autotransformers, these reduce the peak and RMS voltage between live and neutral correctly but in some cases the earth relative voltage may be higher than the live-neutral voltage depending on which way round the autotransformer ends up connected to the mains supply. Autotransformers can also fail in ways that damage the equipment.
• Isolating transformers, these are the safest option but also the bulkiest, heaviest and often most expensive.

I will assume the product you are referring to is

http://www.bestbuy.com/site/insignia-all-in-one-travel-adapter-converter-black/2431031.p?id=1219540952234&skuId=2431031

Looking at the picture (it's two damn small to be a transformer of that power rating) and reading the 1-star reviews I'm pretty convinced that this device is a thyrister/triac based converter and hence is not safe for use with single voltage electronic devices.

In general I would consider voltage conversion as a last resort option. If we are talking about a 360 or XBox One it should be possible to replace the power brick with a 230V model. Unfortunately it doesn't seem that is an option for the original Xbox.