I am a Canadian citizen, entering into a Contract with a US Company (Recruiting firm). The nature of the work requires to travel to Dubai for 6 - 12 months, and some work will also take place in the US (approx 3 months). If I have a TN1 Visa, what kind of visa do I need to travel to and work in Dubai for the US Company?
Even though you will be working for a US company you will still need to get a work permit to work in Dubai. You will also need a residence permit for the period.
Please take a look at the Visas for Dubai page and Work Permits for Dubai.
In most cases employers take care of the visa process for their employees so its a good idea to approach your company's HR department for advice as well.
Good luck.