The UAE's telephone company, Etisalat, offer a pre-paid mobile phone service called Wasel. You can obtain your SIM card immediately from any Etisalat office by presenting your passport and paying a small fee (the page of your passport with your UAE visa on it will be copied).
Cards to add call credit to your mobile phone are available in petrol stations and supermarkets. You can also add credit to your account using the machines provided by Etisalat in large shopping centres. The SIM card is valid for one year only, after which you will need to renew it if you want to continue using the same number.
The Wasel plan also provides international roaming, so you will be able to use your mobile phone outside Dubai.
More information is available on the Etisalat website, in the Travellers > Visitors section.