From the main Terminal 1 at Dubai International Airport (pretty much every international airline uses this except Emirates, which works exclusively from Terminal 3), there are the following hire car companies: Avis (www.avisuae.ae), Budget (www.budget-uae.com), Diamond Lease (www.diamondlease.com), Dollar (www.dollaruae.com), Hertz (www.hertz-uae.com), Europcar (www.europcar-dubai.com), Sixt (www.sixt.com/dubai), Thrifty (www.thriftyuae.com).
All of the companies listed above have offices scattered across the city, in hotels and office blocks. Use their websites or numbers to establish which office is closest to you. Hotels often have a preferred provider, or even rent cars out themselves, so it's worth seeing if you can get a decent offer from reception.
What you need to know about car hire
Generally speaking, for short-term leases, car rental starts at around 180 dirhams per day, and for this you'll probably be getting a Mitsubishi Lancer or Toyota Yaris (which, personally-speaking, I find absolutely fine for the Dubai roads). If you want to compete with the expensive Italian sportsmobiles or vast 4x4 Juggernaughts that litter the city's roads, however, then the sky's the limit in terms of price.
UK licence holders can rent, so long as they're on a tourist visa. If you've got a UAE residency visa, then you'll need to get a UAE licence (which is fairly straightforward, if a little tedious).
If you fancy driving to Oman (which I've just done and is very nice, particularly Muscat, five hours away), you'll need special Omani insurance, which only a few rental companies provide and costs extra. Avis, Budget, Hertz and Thrifty offer such insurance.
To be brutally honest though, unless you've driven in Dubai before, I wouldn't recommend hiring a car for your average holiday. The driving, while not as appalling as, say, Egypt, is still pretty terrible, and the roads are often seen as racing tracks for millionaire boy racers in supercharged sports cars or SUVs, who weave in an out of the traffic at breakneck speeds, without care or concern for anyone around. There's a reason that the Sheikh Zayed Road is one of the most dangerous in the world and you've really got to be on your guard at all times. In my time here I've see crashes far worse than anything on TV. Unfortunately, one 'rude' gesture behind the wheel to an eejit who has just cut you up could land you in serious trouble. Also, it's extremely easy to get yourself lost, with signposts that often seem created solely to confuse and turn-offs that appear out of nowhere. Unless you know the back roads and serious road planning issues off by heart, you could spend a lot of time facing the wrong direction on the wrong road going nowhere near where you want to go. Taxis are cheap and easy to find - go for this instead and save yourself the stress. Read more on my How to get around Dubai page.