First and foremost you have to fulfil these two requirements to be eligible.
1.UAE Resident This is a stamp in your passport stating your status as permanent(a three or one year visa)

2.Citizenship You have to have passport of one of the countries below. Just mere residence in these countries wouldnt make you eligible.

These are countries from which citizens can transfer are given below:

Australia, Austria, Bahrain, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, South Korea, Kuwait, Netherlands, New Zealand, Norway, Oman, Poland, Qatar, Romania, Saudi Arabia, South Africa, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.

There are two more points to be noted here before we get into procedure.

Point One: If you are Canadian
You should get a letter from your consulate in Dubai to verify that your license is genuine. Surely there will be others before you who have gone through the same routine. Just ask your consulate, they will help you out with this.

Point Two:
If you are from Greece, Canada, Turkey, Japan or South Korea
You would need to get it translated and attested from your respective consulates.

Next comes the list of documents you need and fees you must pay:
1.Your Passport (Copy), dont forget to take original with you.
2. Visa Page(Copy) of your Passport
3.Translation (see point 2 above) if its applicable.
4.Letter from Consulate (see point 1 above) if applicable.
5.Two Photographs
6.Eye Test Certificate(See below in Steps)
7.No Objection Letter from your Sponsor.
8.Fee of AED360

Steps to Get Your License.

1.Get an eye test done. This can be done at any hospital, clinic or optician. Give them 2 passport-sized photos and mention that it is for a driving license. They will charge you a small fee.
2.Carry all documents listed above to Dubai Traffic Department in AL Ghusais.
3.Go to Typist counter who for a small fee will do your application in Arabic.
4.Sign your application form and submit documents to the officer
5.Take signed off documents from officer and Pay AED360 to Cashier
6.Submit these to Data Entry Counter
7.You Name will Called Out and your photograph will be taken
8.A few minutes later you will receive your driving license.

Things you must remember after you get your license.

Most Insurance companies do take into account your previous driving experience. So dont forget to bring along any no claims certificates from your home country. Do give it a mention when looking for Car Insurance in Dubai. Because if you will not, it will be assumed you had passed local driving test, hence new driver and licenses which are less than a year in UAE are considered high risk and attract far higher premiums.